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A restricted artinian ring is a commutative ring with an identity in which every proper homomorphic image is artinian. Cohen proved that a commutative ring R is restricted artinian if and only if it is noetherian and every nonzero prime ideal of R is maximal. Facchini and Nazemian called a commutative ring isoartinian if every descending chain of ideals becomes stationary up to isomorphism. We show that every proper homomorphic image of a commutative noetherian ring R is isoartinian if and only if R has one of the following forms: (a) R is a noetherian domain of Krull dimension one which is not a principal ideal domain; (b) R ≅ D 1 × ⋯ × D k × A 1 × ⋯ × A l , where each D i is a principal ideal domain and each A i is an artinian local ring (either k or l may be zero); (c) R is a noetherian ring of Krull dimension one, simple unique minimal prime ideal 픭 , and R / 픭 is a principal ideal domain. As an application of our result, we describe commutative rings whose proper homomorphic images are principal ideal rings. Some relevant examples are provided. [ABSTRACT FROM AUTHOR]

In this paper, we first study zero divisor graphs over finite chain rings. We determine their rank, determinant, and eigenvalues using reduction graphs. Moreover, we extend the work to zero divisor graphs over finite commutative principal ideal rings using a combinatorial method, finding the number of positive eigenvalues and the number of negative eigenvalues, and finding upper and lower bounds for the largest eigenvalue. Finally, we characterize all finite commutative principal ideal rings such that their zero divisor graphs are complete and compute the Wiener index of the zero divisor graphs over finite commutative principal ideal rings. [ABSTRACT FROM AUTHOR]

By the assumption of the theorem and the definition of the direct sum of rings, among these rows there are HT ht is the smallest possible number) that distinguish the elements ( I xb i SB 1 sb ), ..., ( I xb SB k sb i ) of ring I A i I B i , where I x i = I a i SB 1 sb , ..., I a SB n sb i , I x i I A i . Among these rows, by the hypothesis of the theorem and the definition of the direct sum of rings, there are HT ht rows (the minimal possible number) that distinguish these elements. Consider a source of faults HT ht capable of acting on a Boolean function as follows. [Extracted from the article]

Let H m , n be the mn2-dimensional Radford Hopf algebra over an algebraically closed field of characteristic zero. In this paper, we show that any finite dimensional indecomposable H m , n -module is a cyclic module. According to the structure of any indecomposable module under the adjoint action of H m , n , we prove that the Radford Hopf algebra H m , n is a principal ideal ring. Furthermore, we give explicitly the generators of all 28 ideals of 9-dimensional Radford Hopf algebra H 1 , 3 . [ABSTRACT FROM AUTHOR]

Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero. In this paper we show that any finite-dimensional indecomposable H -module is generated by one element. In particular, any indecomposable submodule of H under the adjoint action is generated by a special element of H. Using this result, we show that the Hopf algebra H is a principal ideal ring, i.e., any two-sided ideal of H is generated by one element. As an application, we give explicitly the generators of ideals, primitive ideals, maximal ideals and completely prime ideals of the Taft algebras. [ABSTRACT FROM AUTHOR]

Several different versions of "factoriality" have been defined for commutative rings with zero divisors. We apply semigroup theory to study these notions in the context of a commutative monoid ring R[S], determining necessary and sufficient conditions for R[S] to be various kinds of "unique factorization rings." Our work generalizes Anderson et al.'s results about "unique factorization" in R[X], Gilmer and Parker's characterization of factorial monoid domains, and Hardy and Shores's classification of when R[S] is a principal ideal ring (for S cancellative). Along the way, we determine when R[S] is "restricted cancellative" or satisfies various "(restricted) ideal cancellation laws." [ABSTRACT FROM AUTHOR]

By modifying the existing definition of a compressed zero-divisor graph Γ E (K) , we define a compressed zero-divisor graph Θ (K) of a finite commutative unital ring K , where the compression is performed by means of the associatedness relation (a refinement of the relation used in the definition of Γ E (K)). We prove that this is the best possible compression which induces a functor Θ , and that this functor preserves categorial products (in both directions). We use the structure of Θ (K) to characterize important classes of finite commutative unital rings, such as local rings and principal ideal rings. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the structure of rings over which every essential extension of a direct sum of a family of simple modules is a direct sum of automorphism-invariant modules. We show that if R is a right quotient finite dimensional (q.f.d.) ring satisfying this property, then R is right Noetherian. Also, we show a von Neumann regular (semiregular) ring R with this property is Noetherian. Moreover, we prove that a commutative ring with this property is an Artinian principal ideal ring. [ABSTRACT FROM AUTHOR]

A generalization of Köthe rings is the family of rings whose ideals are direct sums of cyclic modules. These rings were previously studied in the commutative local case. This motivated us to study commutative rings with local dimension whose ideals are direct sums of cyclic modules. First, we obtain an structure theorem for rings of local dimension ω. Then we conclude that, for a ring of local dimension ω , every ideal of R is a direct sum of cyclic modules if and only if every indecomposable ideal of R is cyclic if and only if every maximal ideal of R / Soc (R) is cyclic if and only if every maximal ideal of R is cyclic if and only if R is a principal ideal ring. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative principal ideal ring with unity. In this article, we prove that the zero-divisor graph г(R) is a divisor graph if and only if R is a local ring or it is a product of two local rings with at least one of them having diameter less than 2. We also prove that г(R) is a divisor graph if and only if г(R[x]) is a divisor graph if and only if г(R[[x]]) is a divisor graph. [ABSTRACT FROM AUTHOR]

By two results of Köthe and Cohen-Kaplansky we obtain that 'a commutative ring has the property that every -module is a direct sum of (completely) cyclic modules if and only if is an Artinian principal ideal ring' (an -module is called completely cyclic if each submodule of is cyclic). In this paper, we describe and study commutative rings whose proper ideals are direct sum of completely cyclic modules. It is shown that every proper ideal of a commutative ring is a direct sum of completely cyclic -modules if and only if is a principal ideal ring or is a local ring with maximal ideal such that there is an index set and a set of elements such that with each a simple -module and a principal ideal ring. [ABSTRACT FROM AUTHOR]

Many studies have been conducted to characterize commutative rings whose finitely generated modules are direct sums of cyclic modules (called FGC rings), however, the characterization of noncommutative FGC rings is still an open problem, even for duo rings. We study FGC rings in some special cases, it is shown that a local Noetherian ring R is FGC if and only if R is a principal ideal ring if and only if R is a uniserial ring, and if these assertions hold R is a duo ring. We characterize Noetherian duo FGC rings. In fact, it is shown that a duo ring R is a Noetherian left FGC ring if and only if R is a Noetherian right FGC ring, if and only if R is a principal ideal ring. [ABSTRACT FROM AUTHOR]

LetVbe a finite-dimensional vector space over a fieldk, and letWbe a 1-dimensionalk-vector space. Let ⟨,⟩:V × V → Wbe a symmetric bilinear form. Then ⟨,⟩ is calledanisotropicif for all nonzerov ∈ Vwe have ⟨ v,v ⟩ ≠ 0. Motivated by a problem in algebraic number theory, we give a generalization of the concept ofanisotropyto symmetric bilinear forms on finitely generated modules over artinian principal ideal rings. We will give many equivalent definitions of this concept of anisotropy. One of the definitions shows that a form is anisotropic if and only if certain forms on vector spaces are anisotropic. We will also discuss the concept ofquasi-anisotropyof a symmetric bilinear form, which has no vector space analogue. Finally, we will discuss theradical rootof a symmetric bilinear form, which does not have a vector space analogue either. All three concepts have applications in algebraic number theory. [ABSTRACT FROM AUTHOR]

Abstract: A finite ring R and a weight w on R satisfy the Extension Property if every R-linear w-isometry between two R-linear codes in extends to a monomial transformation of that preserves w. MacWilliams proved that finite fields with the Hamming weight satisfy the Extension Property. It is known that finite Frobenius rings with either the Hamming weight or the homogeneous weight satisfy the Extension Property. Conversely, if a finite ring with the Hamming or homogeneous weight satisfies the Extension Property, then the ring is Frobenius. This paper addresses the question of a characterization of all bi-invariant weights on a finite ring that satisfy the Extension Property. Having solved this question in previous papers for all direct products of finite chain rings and for matrix rings, we have now arrived at a characterization of these weights for finite principal ideal rings, which form a large subclass of the finite Frobenius rings. We do not assume commutativity of the rings in question. [Copyright &y& Elsevier]

This paper investigates the structure of cyclically pure (or C-pure) projective modules. In particular, it is shown that a ringRis left Noetherian if and only if every C-pure projective leftR-module is pure projective. Also, over a left hereditary Noetherian ringR, a leftR-moduleMis C-pure projective if and only ifM = N⊕P, whereNis a direct sum of cyclic modules andPis a projective leftR-module. The relationship C-purity with purity and RD-purity are also studied. It is shown that ifRis a local duo-ring, then the C-pure projective leftR-modules and the pure projective leftR-modules coincide if and only ifRis a principal ideal ring. IfRis a left perfect duo-ring, then the C-pure projective leftR-modules and the pure projective leftR-modules coincide if and only ifRis left Köthe (i.e., every leftR-module is a direct sum of cyclic modules). Also, it is shown that for a ringR, if every C-pure projective leftR-module is RD-projective, thenRis left Noetherian, every p-injective leftR-module is injective and every p-flat rightR-module is flat. Finally, it is shown that ifRis a left p.p-ring and every C-pure projective leftR-module is RD-projective, thenRis left Noetherian hereditary. The converse is also true whenRis commutative, but it does not hold in the noncommutative case. [ABSTRACT FROM PUBLISHER]

Abstract: Let be a commutative, local, and principal ideal ring with maximal ideal and residue class field . Suppose that every element of is square. Then the problem of classifying arbitrary symmetric matrices over by congruence naturally reduces, and is actually equivalent to, the problem of classifying invertible symmetric matrices over by congruence. [Copyright &y& Elsevier]

Abstract: For R an artinian ring and G a group, we characterize when RG is a principal ideal ring. In the case when G is finite (and R artinian), this yields a characterization of when RG is a left and right morphic ring. This extends work done by Passman, Sehgal and Fisher on principal ideal group rings when the coefficient ring is a field, and work of Chen, Li, and Zhou on morphic group rings. [Copyright &y& Elsevier]

A ring R is called left morphic if R/Ra ≅ l(a) for each a ∈ R, equivalently if there exists b ∈ R such that Ra = l(b) and l(a) = Rb. In this paper, we ask only that b and c exist such that Ra = l(b) and l(a) = Rc, and call R left quasi-morphic if this happens for every element a of R. This class of rings contains the regular rings and the left morphic rings, and it is shown that finite intersections of principal left ideals in such a ring are again principal. It is further proved that if R is quasi-morphic (left and right), then R is a Bézout ring and has the ACC on principal left ideals if and only if it is an artinian principal ideal ring. [ABSTRACT FROM AUTHOR]

Abstract: We characterize rings over which every projective module is a direct sum of finitely generated modules, and give various examples of rings with and without this property. [Copyright &y& Elsevier]

Abstract: We show that repeated-root cyclic codes over a finite chain ring are in general not principally generated. Repeated-root negacyclic codes are principally generated if the ring is a Galois ring with characteristic a power of 2. For any other finite chain ring they are in general not principally generated. We also prove results on the structure, cardinality and Hamming distance of repeated-root cyclic and negacyclic codes over a finite chain ring. [Copyright &y& Elsevier]

Let R be a ring. An R-module M is said to satisfy property (S) if every subjective endomorphism of M is an automorphism. Let FR be the class of finitely generated R-module and SR be the class of R-module satisfying property (S). In 1969 W. V. Vasconcelos proved that for every commutative ring R, FR ⊆ SR. In general this inclusion is strict. In this note we characterize commutative ring R for which FR = SR. [ABSTRACT FROM AUTHOR]

By using the results about polynomial matrix in the system and control theory, this paper gives some discussions about the algebraic properties of polynomial matrices. The obtained main results include that the ring of nxn polynomial matrices is a principal ideal, and principal one-sided ideal ring. [ABSTRACT FROM AUTHOR]