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The set is a sub-f-ring of RL, that is, the ring of all continuous real-valued functions on a completely regular frame L. The main purpose of this paper is to continue our investigation begun in [3] of extending ring-theoretic properties in RL to the context of completely regular frames by replacing the ring RL with the ring Cc(L) to the context of zero-dimensional frames. We show that a frame L is a CP-frame if and only if Cc(L) is a regular ring if and only if every ideal of Cc(L) is pure if and only if Cc(L) is an Artin-Rees ring if and only if every ideal of Cc(L) with the Artin-Rees property is an Artin-Rees ideal if and only if the factor ring Cc(L)=hi is an Artin-Rees ring for any 2 Cc(L) if and only if every minimal prime ideal of Cc(L) is an Artin-Rees ideal. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity and M an R-bimodule. We introduce the generalization of r-clean rings called left r-clean R-bimodules, defined without their endomor-phism rings. An R-bimodule M is said to be left r-clean if each element is the sum of a left idempotent and a left regular element of M. We present some properties of the left r-clean R-bimodule. At the end of this paper, we give the suficient and necessary condition for an R-bimodule to form a left r-clean R-bimodule. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with non-zero identity, S ⊆ R a multiplicatively closed subset of R and M a unital R -module. In this paper, we introduce the concepts of S -almost semiprime submodules and S -weakly semiprime submodules as a generalization of S -semiprime submodules. We investigate some basic properties of S -almost semiprime and S -weakly semiprime submodules and give some characterizations of them, especially, for (finitely generated faithful) multiplication modules. [ABSTRACT FROM AUTHOR]

The notion of a power MR-group, where R is an arbitrary associative ring with unity, was introduced by R. Lyndon. A. G. Myasnikov and V. N. Remeslennikov gave a more precise definition of an R-group by introducing an additional axiom. In particular, the new notion of a power MR-group is a direct generalization of the notion of an R-module to the case of noncommutative groups. In the present paper, central series and series of commutants in MR-groups are introduced. Three versions of the definition of nilpotent power MR-groups of step n are discussed. We prove that all these definitions are equivalent for n = 1, 2. The question on the coincidence of these notions for n > 2 remains open. Moreover, we prove that the tensor completion of a 2-step nilpotent MR-group is 2-step nilpotent. [ABSTRACT FROM AUTHOR]

In this paper, we study third power associative rings satisfying the identity a (b c) = b (c a). We prove that third power associative rings satisfying the identity a (b c) = b (c a) with characteristics ≠ 2 are power associative and associators generate nilpotent ideal with index of nilpotency ≤ 2 . Furthermore, we prove that third power associative rings satisfying the identity a (b c) = b (c a) with commutators contained in the left nucleus and characteristics ≠ 2 , 3 are associative and commutative of degree five. [ABSTRACT FROM AUTHOR]

Let g be an algebra over K with a bilinear operation [ ⋅ , ⋅ ] : g × g → g not necessarily associative. For A ⊆ g , let A k be the set of elements of g written combining k elements of A via + and [ ⋅ , ⋅ ]. We show a "sum-bracket theorem" for simple Lie algebras over K of the form g = sl n , so n , sp 2 n , e 6 , e 7 , e 8 , f 4 , g 2 : if char (K) is not too small, we have growth of the form | A k | ≥ | A | 1 + ε for all generating symmetric sets A away from subfields of K. Over F p in particular, we have a diameter bound matching the best analogous bounds for groups of Lie type [2]. As an independent intermediate result, we prove also an estimate of the form | A ∩ V | ≤ | A k | dim ⁡ (V) / dim ⁡ (g) for linear affine subspaces V of g. This estimate is valid for all simple algebras, and k is especially small for a large class of them including associative, Lie, and Mal'cev algebras, and Lie superalgebras. [ABSTRACT FROM AUTHOR]

The paper deals with new specific constructions of indecomposable torsion-free abelian groups of rank two and nonzero rings on them. They illustrate purely theoretical results and complement quite rare examples obtained during the classical as well as recent research of additive groups of rings. The presented results concerning the homogeneous groups remain true for groups of any finite nonzero rank. Moreover, the paper contains a construction of a torsion-free indecomposable abelian group of an arbitrary finite rank greater than two supporting an associative, but not commutative ring, as well as a ring which is neither associative nor commutative. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the structure of irreducible modules in the variety ℳ of commutative power-associative nilalgebras of nilindex ≤ 4. If A ∈ ℳ with dimension at most 5, then we prove that A 2 is contained in the annihilator of every irreducible A -module in the variety ℳ. Also, we consider the enveloping algebra of an algebra A in the variety ℳ and we obtain a new example of a commutative power-associative non-nilpotent nilalgebra of dimension 9. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with unity, X be a finite group, H be a subgroup of X, and G be a set of left coset representatives for the left action of H on X. In this article, we introduce two different ways to put R into a non-trivial G -weak graded ring that is a ring graded by the set G which is defined with a binary operation ∗ and satisfying an algebraic structure with specific properties. The first one is by choosing a subset S of G such that S is a group under the ∗ operation and putting R t = 0 for all t ∈ G and t ∉ S . The second way, which is the most important, is induced by combining the operation ∗ defined on G and the coaction ◁ of H on G. Many examples are provided. [ABSTRACT FROM AUTHOR]

Let R be an associative ring and let s ≥ 1 be a fixed integer. An additive map h on R is called a homoderivation if h (x y) = h (x) h (y) + h (x) y + x h (y) holds for all x , y ∈ R. In 1978, Herstein proved that a prime ring R of char (R) ≠ 2 is commutative if there is a nonzero derivation d of R such that [ d (x) , d (y) ] = 0 for all x , y ∈ R . The main objective of this paper is to prove the above mentioned result for homoderivations with nilpotency 's' in prime rings. Moreover, we prove that if a prime ring admits homoderivations h1 and h2 such that h 1 ° h 2 = h 2 ° h 1 and h 1 s 1 ° h 2 s 2 = 0 for positive integers s1 and s2, then at least one of the homoderivations must be nilpotent. [ABSTRACT FROM AUTHOR]

Let R and Rφ be associative rings with isomorphic subring lattices, and φ be a lattice isomorphism (or else a projection) of the ring R onto the ring Rφ. We call Rφ the projective image of a ring R and call R itself the projective preimage of a ring Rφ. The main result of the first part of the paper is Theorem 5, which proves that the projective image Rφ of a one-generated finite p-ring R is also one-generated if Rφ at the same time is itself a p-ring. In the second part, we continue studying projections of matrix rings. The main result of this part is Theorems 6 and 7, which prove that if R = Mn(K) is the ring of all square matrices of order n over a finite ring K with identity, and φ is a projection of the ring R onto the ring Rφ, then Rφ = Mn(K′), where K′ is a ring with identity, lattice-isomorphic to the ring K. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the notions uniformly S-flat, briefly u-S-flat, modules and dimensions. We introduce and study the notions of weak u-S-flat modules. An R-module M is said to be weak u-S-flat if Tor1R(R/I,M) is u-S-torsion for any ideal I of R. This new class of modules will be used to characterize u-S-von Neumann regular rings. Hence, we introduce the weak u-S-flat dimensions of modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we will extend some results on the commutativity of Jordan ideals proved in [8] and [5]. However, instead of left generalized derivations, we will consider generalized left derivations, which are sufficient to obtain good results with respect to the structure of near-rings. We will also show that the conditions imposed in the paper cannot be removed. [ABSTRACT FROM AUTHOR]

An associative ring R with identity is a weakly g(x)-invo-clean ring if every r 2 R can be written as r = v + s or r = v + s, where v2 = 1 and s is a root of g(x) 2 C(R)[x]. It is proved that an associtive ring R is weakly invo-clean and 22 = 1 if and only if R is weakly (x2+2x)-invo-clean if and only if R is weakly (x2+2x)-invo-clean. [ABSTRACT FROM AUTHOR]

In the two subvarieties of commutative not necessarily associative rings, and 2-divisible not necessarily associative rings, we show that the sixty Bol-Moufang identities determine exactly five subvarieties in each case, and we completely determine all inclusions and necessary counterexamples. The historically important nonassociative rings of octonions and sedenions may be used as counterexamples in our classification. We utilize both associator and linear forms of each Bol-Moufang identity. The results are analogous to the classification of varieties of loops or quasigroups of Bol-Moufang type (Phillips, Vojtechovský-2005). [ABSTRACT FROM AUTHOR]

Let O(k) be the octonion Cayley-Dickson algebra over a commutative associative ring κ with 1. Let G(k) be the Moufang loop of invertible elements of O(k). Let H be a class of groups such that a group G is a member of H if and only if G satisfies the following three conditions: (a) G is not class-2 nilpotent. (b) G has a proper class-2 nilpotent subgroup. (c) G is not isomorphic to any subgroup of the group GL2(F) for any field F. The theorem proved in the paper states that if κ is an integral domain with 1+1 6= 0, then G(k) does not contain any subloop isomorphic to a group of class H, while if κ is an integral domain such that 1+1 = 0, then G(k) contains no subloop isomorphic to a class-2 nilpotent group at all. [ABSTRACT FROM AUTHOR]

Let ℂ q be a rational quantum torus associated with the matrix q. Let Der (ℂ q) be the Lie algebra of derivations of ℂ q . In this paper, we consider the Lie algebra (ℂ q ⋊ Der (ℂ q)) ⊗ B , where B is a commutative associative unital algebra over ℂ and classify its irreducible modules with finite-dimensional weight spaces. [ABSTRACT FROM AUTHOR]

Tensor product of highest weight modules and intermediate modules for Virasoro algebra have been studied around 1997. Since then the irreducibility problem for tensor product of modules is open. We consider the loop-Virasoro algebra V i r ⊗ B , where Vir is the Virasoro algebra and B is a commutative associative unital algebra over ℂ. In this paper, we study the irreducibility problem for the tensor product of highest weight modules and intermediate modules for Vir ⊗ B. Finally we found out a necessary and sufficient condition for such modules to be isomorphic. [ABSTRACT FROM AUTHOR]

This paper is devoted to study of local and 2-local derivations on octonion algebras. We shall give a general form of local derivations on the real octonion algebra ℝ . This description implies that the space of all local derivations on ℝ when equipped with Lie bracket is isomorphic to the Lie algebra 7 (ℝ) of all real skew-symmetric 7 × 7 -matrices. We also consider 2 -local derivations on an octonion algebra over an algebraically closed field of characteristic zero and prove that every 2 -local derivation on is a derivation. Further, we apply these results to similar problems for the simple seven-dimensional Malcev algebra. As a corollary, we obtain that the real octonion algebra ℝ and Malcev algebra M 7 (ℝ) are simple non-associative algebras which admit pure local derivations, that is, local derivations which are not derivation. [ABSTRACT FROM AUTHOR]

Let R be an associative ring and let n be a non-negative integer. In this paper, we consider n-super finitely presented, Gorenstein n-weak injective and Gorenstein n-weak flat modules under change of rings. For an excellent extension S ≥ R, we show that the Gorenstein n-super finitely presented dimensions (resp., weak Gorenstein n-super finitely presented dimensions) of rings R and S coincide. [ABSTRACT FROM AUTHOR]

Let R be a ring with an endomorphism α. A ring R is a skew power series McCoy ring if whenever any non-zero power series... We generalized the results of [2] and investigate the relations between the skew power series ring and the standard ring-theoretic properties. Moreover, we obtain some characterizations for skew power series ring R[[x; α]], to be McCoy, zip, strongly AB and has Property (A). [ABSTRACT FROM AUTHOR]

Groups of infinite unitriangular matrices over associative unitary rings are considered. These groups naturally act on infinite dimensional free modules over underlying rings. They are profinite in case underlying rings are finite. Inspired by their connection with groups defined by finite automata the problem to construct faithful representations of free products of groups by banded infinite unitriangular matrices is considered. For arbitrary prime p a sufficient conditions on a finite set of banded infinite unitriangular matrices over unitary associative rings of characteristic p under which they generate the free product of cyclic p-groups is given. The conditions are based on certain properties of the actions on finite dimensional free modules over underlying rings. It is shown that these conditions are satisfied. For arbitrary free product of finite number of cyclic p-groups constructive examples of the sets of infinite unitriangular matrices over unitary associative rings of characteristic p that generate given free product are presented. These infinite matrices are constructed from finite dimensional ones that are nilpotent Jordan blocks. A few open questions concerning properties of presented examples and other types of faithful representations are formulated. [ABSTRACT FROM AUTHOR]

In this paper, we study the deformation theory of associative conformal algebras and conformal bimodules. Using the O -operator on bimodules over associative conformal algebras, we introduce the notion of O N -structure on bimodules over associative conformal algebras and consider the relationship between it and the compatibility of O -operators, the PN-structure, the Ω N -structure and the P Ω -structure on associative conformal algebra. We show that a solution of the strong Maurer-Cartan equation of an associative conformal twilled algebra gives rise to an O N -structure, and an O N -structure satisfying certain conditions gives a solution of the strong Maurer-Cartan equation of an associative conformal twilled algebra. [ABSTRACT FROM AUTHOR]

Let R be an associative ring. Then R is said to be coverable provided R is the union of its proper subrings (which we do not require to be unital even if R is so). One verifies easily that R is coverable if and only if R is not generated as a ring by a single element. In case R can be expressed as the union of a finite number of proper subrings, the least such number is called the covering number of R. Covering numbers of rings have been studied in a series of recent papers. The purpose of this note is to study rings which can be covered by a countable collection of subrings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of soft quasi-ideal, soft minimal quasi-ideal, soft left (resp. right) N-subgroup and soft invariant subnear-ring of a soft near-ring with the help of soft set theory established by Molodtsov. We also introduce the concepts of soft zero-symmetric near-ring, soft constant near- ring, soft near-field and soft Q- simple near-ring over a near-ring. We investigate the properties of these notions with illustrative examples. We obtain the characterizations of soft quasi-ideals of a soft (zero-symmetric) near-ring and soft nearfields over a near-ring. [ABSTRACT FROM AUTHOR]

In this paper, the author gives a description of the derivations of U T M n (R) , the ring of upper triangular matrices over an associative ring R with identity. The main result states that if D is an arbitrary derivation of the ring U T M n (R) and A ∈ U T M n (R) , then there are matrices, such that the derivative D (A) is a linear combination of the values of well-known derivations of these matrices. [ABSTRACT FROM AUTHOR]

We present and study the concept of m -periodic Gorenstein objects relative to a pair (A , B) of classes of objects in an abelian category, as a generalization of m -strongly Gorenstein projective modules over associative rings. We prove several properties when (A , B) satisfies certain homological conditions, like for instance when (A , B) is a GP-admissible pair. Connections to Gorenstein objects and Gorenstein homological dimensions relative to these pairs are also established. [ABSTRACT FROM AUTHOR]

We return to the work of Banaschewski and extract from it a theorem of Fuchs, Heinzer, and Olberding. As an application of Fuchs-Heinzer-Olberding's theorem, we generalize a result of Gillman and Kohls. We study pseudo-irreducible ideals and show that every ideal of a pm-ring is the (not necessarily finite) intersection of pairwise comaximal pseudo-irreducible ideals. After some general results, the article focuses on primal and pseudo-irreducible ideals in rings of continuous functions. We determine when every pseudo-irreducible ideal of C(X) is primal. We give a characterization of spaces X for which every Op is a primal ideal of C(X). [ABSTRACT FROM AUTHOR]

Suppose is an alternative ring containing a non-trivial idempotent and χ m be a mapping from into itself. In this paper, we study the Jordan n-higher derivations on alternative rings and prove that under some mild conditions every multiplicative Jordan n-higher derivations on is additive. It is to be noted that the similarity to the notions is only in its written form, but not in its theoretical structures, because the Pierce decomposition used in the results for alternative rings is the generalization of the Pierce decomposition for associative rings. [ABSTRACT FROM AUTHOR]

For a matrix incidence algebra over a field, there is a full description of its generating subsets. We generalise this result to the case of matrix incidence rings over an arbitrary unital associative ring. Generating subsets of the following type are under consideration: if one adds all scalar matrices to the subset, then this greater set will generate the incidence ring. We obtain the full characterisation of such subsets. Moreover, their minimum cardinality is calculated precisely when the ring of coefficients is simple or semilocal. In order to obtain these results we introduce the graph of full differences of a ring and study its clique number. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with unity. Two new concepts namely "e-small" and "e-supplement" in R are introduced and many of its properties are discussed in this paper. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity. Ring R is considered clean if each element is the sum of an idempotent element and a unit element. A clean ring is generalized to an r-clean ring by generalizing the unit elements to regular. A ring R is said to be r-clean if each element is the sum of an idempotent element and a regular element. We introduce the notion of the r-clean ideal, a natural generalization of the clean ideal. Furthermore, we present some properties of the r-clean ideal and offer several sufficient and necessary conditions for a clean ideal to be r-clean. [ABSTRACT FROM AUTHOR]

We show that the condition of being categorical in a tail of cardinals can be characterized algebraically for several classes of modules. Theorem 0.1 Assume R is an associative ring with unity. (1) The class of locally pure-injective R-modules is λ-categorical in all λ > card (R) + ℵ 0 if and only if R ≅ M n (D) for D a division ring and n ≥ 1. (2) The class of flat R-modules is λ-categorical in all λ > card (R) + ℵ 0 if and only if R ≅ M n (k) for k a local ring such that its maximal ideal is left T-nilpotent and n ≥ 1. (3) Assume R is a commutative ring. The class of absolutely pure R-modules is λ-categorical in all λ > card (R) + ℵ 0 if and only if R is a local artinian ring. We show that in the above results it is enough to assume λ -categoricity in some big cardinal λ. This shows that Shelah's Categoricity Conjecture holds for the class of locally pure-injective modules, flat modules and absolutely pure modules. These classes are not first-order axiomatizable for arbitrary rings. We provide rings such that the class of flat modules is categorical in a tail of cardinals but it is not first-order axiomatizable. [ABSTRACT FROM AUTHOR]

An element a of a ring R is called a quasi-idempotent if a 2 = k a for some central unit k of R , or equivalently, a = k e , where k is a central unit and e is an idempotent of R. A ring R is called a quasi-Boolean ring if every element of R is quasi-idempotent. A ring R is called (strongly) quasi-clean if each of its elements is a sum of a quasi-idempotent and a unit (that commute). These rings are shown to be a natural generalization of the clean rings and strongly clean rings. An extensive study of (strongly) quasi-clean rings is conducted. The abundant examples of (strongly) quasi-clean rings state that the class of (strongly) quasi-clean rings is very larger than the class of (strongly) clean rings. We prove that an indecomposable commutative semilocal ring is quasi-clean if and only if it is local or R has no image isomorphic to ℤ 2 ; For an indecomposable commutative semilocal ring R with at least two maximal ideals, n (R) (n ≥ 2) is strongly quasi-clean if and only if n (R) is quasi-clean if and only if min { | R / m | , m is a maximal ideal of R } > n + 1. For a prime p and a positive integer n ≥ 2 , n (ℤ (p)) is strongly quasi-clean if and only if p > n. Some open questions are also posed. [ABSTRACT FROM AUTHOR]

With this paper we present an extension of our recent ISSAC paper about computations of Gröbner(-Shirshov) bases over free associative algebras Z 〈 X 〉. We present all the needed proofs in details, add a part on the direct treatment of the ring Z / m Z as well as new examples and applications to e.g. Iwahori-Hecke algebras. The extension of Gröbner bases concept from polynomial algebras over fields to polynomial rings over rings allows to tackle numerous applications, both of theoretical and of practical importance. Gröbner and Gröbner-Shirshov bases can be defined for various non-commutative and even non-associative algebraic structures. We study the case of associative rings and aim at free algebras over principal ideal rings. We concentrate on the case of commutative coefficient rings without zero divisors (i.e. a domain). Even working over Z allows one to do computations, which can be treated as universal for fields of arbitrary characteristic. By using the systematic approach, we revisit the theory and present the algorithms in the implementable form. We show drastic differences in the behaviour of Gröbner bases between free algebras and algebras, which are close to commutative. Even the process of the formation of critical pairs has to be reengineered, together with implementing the criteria for their quick discarding. We present an implementation of algorithms in the Singular subsystem called Letterplace , which internally uses Letterplace techniques (and Letterplace Gröbner bases), due to La Scala and Levandovskyy. Interesting examples and applications accompany our presentation. [ABSTRACT FROM AUTHOR]

Product theorems are available for associative algebras over a scalar ring containing as well as for such algebras of rank 3. We prove the exact analog of a product theorem for alternative algebras of rank 3 and describe the identities in three variables for the alternative algebras Lie nilpotent of a given degree. We also prove an analog of a product theorem in a general form without any restriction on the algebra rank and show that there is no exact analog of a product theorem for alternative algebras. The connection between the concepts of the Lie and strong Lie nilpotency is studied for a given algebra of rank 3 and its multiplication algebra. [ABSTRACT FROM AUTHOR]

We study some properties of the compressed zero-divisor graph of a finite ring and the partially compressed zero-divisor graph of a finite nilpotent ring. In particular, we describe all nilpotent finite rings whose compressed zero-divisor graphs are some complete graphs with loops. Furthermore, we introduce the notion of partially compressed zero-divisor graph for the nilpotent rings and study the properties of the group. Also, we describe the finite rings whose compressed zero-divisor graph contains a bridge. [ABSTRACT FROM AUTHOR]

A commutative ring R is said to satisfy acc on d-annihilators if for every sequence (a k) k ∈ ℕ of elements of R, the sequence ann (a 1) ⊆ ann (a 1 a 2) ⊆ ⋯ is stationary. In this paper we extend the notion of rings with acc on d-annihilators by introducing the concept of rings with S-acc on d-annihilators, where S is a multiplicative set. Let R be a commutative ring and S a multiplicative subset of R. We say that R satisfies S-acc on d-annihilators if for every sequence (a k) k ∈ ℕ of elements of R, the sequence ann (a 1) ⊆ ann (a 1 a 2) ⊆ ⋯ is S-stationary, that is, there exist a positive integer n and an s ∈ S such that for each k ≥ n , s ( ann R (a 1 a 2 ⋯ a k)) ⊆ ann R (a 1 a 2 ⋯ a n). We give equivalent conditions for the power series (respectively, polynomial) ring over an Armendariz ring to satisfy S-acc on d-annihilators. We also study serval properties of rings satisfying S-acc on d-annihilators. The concept of the amalgamated duplication of R along an ideal I, R ⋈ I is studied. [ABSTRACT FROM AUTHOR]

Let R be a G graded commutative ring and M be a G-graded R-module. The set of all graded second submodules of M is denoted by S pecsG(M); and it is called the graded second spectrum of M. We discuss graded rings with Noetherian graded prime spectrum. In addition, we introduce the notion of the graded Zariski socle of graded submodules and explore their properties. We also investigate S pecsG(M) with the Zariski topology from the viewpoint of being a Noetherian space. [ABSTRACT FROM AUTHOR]

Let R be a unitary associative algebra over a field. We call an algebra A a generalized R-algebra when A is endowed with an R -module action with the property that, for each r ∈ R , there exists finitely many elements r + = (r 1 + , r 2 +) ∈ R 2 and r − = (r 1 − , r 2 −) ∈ R 2 such that, for all a 1 , a 2 ∈ A , r ⋅ (a 1 a 2) = ∑ r + (r 1 + ⋅ a 1) (r 2 + ⋅ a 2) + ∑ r − (r 2 − ⋅ a 2) (r 1 − ⋅ a 1). Suppose an associative generalized R -algebra A satisfies an identical relation of the form x 1 ⋯ x d − ∑ 1 ≠ σ ∈ S d ∑ r (r 1 ⋅ x σ (1)) ⋯ (r d ⋅ x σ (d)) ≡ 0 , where S d denotes the symmetric group of degree d and the inner sum runs over finitely many r = (r 1 , ... , r d) ∈ R d. We prove: if the algebra of endomorphisms on A defined by the action of R is m -dimensional, then A satisfies a classical polynomial identity of degree bounded by an explicit function of d and m only. We also prove the analogous result holds when A is a Lie algebra, thus extending a collection of results in associative and Lie PI-theory. [ABSTRACT FROM AUTHOR]

In this paper, we modify the result of M. N. Daif [1] on multiplicative derivations in rings. He showed that the multiplicative derivation is additive by imposing certain conditions on the ring <. Here, we have proved the above result with lesser conditions than M. N. Daif for getting multiplicative derivation to be additive. [ABSTRACT FROM AUTHOR]

The main achievement of this paper is a geometric characterisation of certain subvarieties of the Cartan variety 퓔6(핂) over an arbitrary field 핂. The characterised varieties arise as Veronese representations of certain ring projective planes over quadratic subalgebras of the split octonions 핆' over 핂 (among which the sextonions, a 6-dimensional non-associative algebra). We describe how these varieties are linked to the Freudenthal–Tits magic square, and discuss how they would even fit in, when also allowing the sextonions and other "degenerate composition algebras" as the algebras used to construct the square. [ABSTRACT FROM AUTHOR]