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Let f n ( k ) be the number of subrings of index k in Z n . We show that results of Brakenhoff imply a lower bound for the asymptotic growth of subrings in Z n , improving upon lower bounds given by Kaplan, Marcinek, and Takloo-Bighash. Further, we prove two new lower bounds for f n ( p e ) when e ≥ n − 1 . Using these bounds, we study the divergence of the subring zeta function of Z n and its local factors. Lastly, we apply these results to the problem of counting orders in a number field.

Recursive MDS matrices are used for the design of linear diffusion layers in lightweight cryptographic applications. Most of the works on the construction of recursive MDS matrices either consider matrices over finite fields or block matrices over G L ( m , F 2 ) . In the first case, there have been works on the direct construction of recursive MDS matrices. The latter case is hard to deal with because of its non-commutative nature. There has not been any serious attempt to look for recursive MDS matrices over finite commutative rings, in particular over local rings of even characteristic. In this work, we present several methods for the construction of recursive MDS companion matrices over finite commutative rings. The main tools are the simple expressions for the determinant of (generalized) Vandermonde and linearized matrices. We show that the determinant of a linearized matrix over a finite commutative ring of prime characteristic can be expressed in a simple form. We discuss a technique called subring construction with which MDS matrices over product rings can be constructed using MDS matrices over subrings. We give a few examples of recursive MDS companion matrices over local rings of even characteristic. We also discuss some results on the nonexistence of recursive MDS matrices over certain rings for some parameter choices.

Let $\mathrm{k}_{0}$ be a field and $W$ a finite orthogonal reflection group over $\mathrm{k}_{0}$. We prove Serre's splitting principle for cohomological invariants of $W$ with values in Rost's cycle modules (over $\mathrm{k}_{0}$) if the characteristic of $\mathrm{k}_{0}$ is coprime to $|W|$. We then show that this principle for such groups holds also for Witt- and Milnor-Witt $K$-theory invariants., 20 pages

In this paper, we study the invariant theory of quadratic Poisson algebras. Let G be a finite group of the graded Poisson automorphisms of a quadratic Poisson algebra A. When the Poisson bracket of A is skew-symmetric, a Poisson version of the Shephard-Todd-Chevalley theorem is proved stating that the fixed Poisson subring A^G is skew-symmetric if and only if G is generated by reflections. For many other well-known families of quadratic Poisson algebras, we show that G contains limited or even no reflections. This kind of Poisson rigidity result ensures that the corresponding fixed Poisson subring A^G is not isomorphic to A as Poisson algebras unless G is trivial., Comment: Small revisions throughout. Remark 4.13 added courtesy of Akaki Tikaradze

Consider the universal gate set for quantum computing consisting of the gates X, CX, CCX, omega^dagger H, and S. All of these gates have matrix entries in the ring Z[1/2,i], the smallest subring of the complex numbers containing 1/2 and i. Amy, Glaudell, and Ross proved the converse, i.e., any unitary matrix with entries in Z[1/2,i] can be realized by a quantum circuit over the above gate set using at most one ancilla. In this paper, we give a finite presentation by generators and relations of U_n(Z[1/2,i]), the group of unitary nxn-matrices with entries in Z[1/2,i]., Comment: In Proceedings QPL 2021, arXiv:2109.04886

Suppose that $G$ is a finite group and $k$ is a field of characteristic $p>0$. We consider the complete cohomology ring $\mathcal{E}_M^* = \sum_{n \in \mathbb{Z}} \widehat{Ext}^n_{kG}(M,M)$. We show that the ring has two distinguished ideals $I^* \subseteq J^* \subseteq \mathcal{E}_M^*$ such that $I^*$ is bounded above in degrees, $\mathcal{E}_M^*/J^*$ is bounded below in degree and $J^*/I^*$ is eventually periodic with terms of bounded dimension. We prove that if $M$ is neither projective nor periodic, then the subring of all elements in negative degrees in $\mathcal{E}_M^*$ is a nilpotent algebra., 15 pages, The Version of Record of this article is published in Beitr\"age zur Algebra und Geometrie and is available on line at https://doi.org/10.1007/s13366-021-00595-y

Let D be a division ring with involution ⋆ and S the set of all symmetric elements of D. Assume that the center F of D is uncountable and K is a division subring of D containing F. The main aim of this note is to show that S is right algebraic over K if and only if so is D. This result allows us to construct an example of division rings K ⊂ D such that D is right algebraic but not left algebraic over K.

Let $S$ be an unramified regular local ring of mixed characteristic two and $R$ the integral closure of $S$ in a biquadratic extension of its quotient field obtained by adjoining roots of sufficiently general square free elements $f,g\in S$. Let $S^2$ denote the subring of $S$ obtained by lifting to $S$ the image of the Frobenius map on $S/2S$. When at least one of $f,g\in S^2$, we characterize the Cohen-Macaulayness of $R$ and show that $R$ admits a birational small Cohen-Macaulay module. It is noted that $R$ is not automatically Cohen-Macaulay in case $f,g\in S^2$ or if $f,g\notin S^2$., Comment: Final version, to appear in Journal of Algebra; minor changes, unabbreviated title, updated references

We discuss when an order in a Dedekind domain $$R$$ is equal to $$R$$ (is the maximal order in $$R$$ ). Every order in $$R$$ is a subring of $$R$$ . This fact implies the existence of natural homomorphisms between objects related to orders such that the group of Cartier divisors, the Picard group, the group of Weil divisors, the Chow group and the Witt ring of an order. We examine the maximality of an order in $$R$$ in the context of such natural homomorphisms. In [8], we discuss when an order $$\mathcal{O}$$ in $$R$$ is equal to $$R$$ on the assumption that either the Picard group of $$R$$ or the Picard group of $$\mathcal{O}$$ is a torsion group. In this paper, we abandon this assumption. We formulate equivalent conditions for the maximality of $$\mathcal{O}$$ for any Dedekind domain $$R$$ and any order $$\mathcal{O}$$ in $$R$$ .

Let n be an integer greater than 1 and let x 1 , … , x n be indeterminates over a countable field k. In this paper, we employ techniques of Heitmann and Nagata to show there exists an uncountable regular local ring S between the localized polynomial ring k [ x 1 , … , x n ] ( x 1 , … , x n ) and the power series ring k [ [ x 1 , … , x n ] ] such that the prime ideal spectrum of S is homeomorphic to the prime ideal spectrum of k [ x 1 , … , x n ] ( x 1 , … , x n ) as topological spaces with the Zariski topology ( Theorem 3.17 ). Thus S is a local n-dimensional Noetherian domain and the cardinality of the set of prime ideals of S is strictly less than the cardinality of S. We also show that every Noetherian ring A with infinitely many prime ideals has a Noetherian subring B such that the prime ideal spectrum of B is homeomorphic to the prime ideal spectrum of A and the cardinality of the set of prime ideals of B equals the cardinality of B.

Let R and S be commutative rings with identity, f:R→S a ring homomorphism and J an ideal of S. Then, the subring R⋈fJ:={(a,f(a)+j)|a∈R and j∈J} of R × S is called the amalgamation of R with S along...

Let D be a principal ideal domain (PID), I be an ideal of D, and X be an indeterminate over D. Let [D;I][X] be the subring of the power series ring D[​[X]​] consisting of all power series f=∑i=0∞ai...

Let R be a commutative ring and Ri.c(R) denotes the set of all integrally closed maximal subrings of R. It is shown that if R is a non-field G-domain, then there exists S ∈ Xi.c(R) with (S : |R) = 0. If K is an algebraically closed field which is not absolutely algebraic, then we prove that the polynomial ring K [X] has an integrally closed maximal subring with zero conductor too; a characterization of integrally closed maximal subrings of K [X] with (non-)zero conductor is given. It is observed that, an integrally closed maximal subrings S of K [X] is a principal ideal domain (PID) if and only if M = Sq for some q-1 ∈K \ S, where M is the crucial maximal idea of the extension S ⊆ K [X]. We show that if f (X,Y ) is an irreducible polynomial in K [X,Y ], then there exists an integrally closed maximal subring S of K [X, Y ] with (S : K [X, Y ]) = f(X,Y )K [X,Y ]. It is proved that, if R is a ring and S (I) = {T ∈ Xi:c(R) |I ⊆ T}, where I is an ideal of R, then S := {S(I) | I is an ideal of R} is a topology for closed sets on Xi:c(R). We show that this space has similar properties such as those one in the Zariski spaces on Spec(R) or Kn (the affine space). In particular, if K is a field which is not algebraic over its prime subring, then Xi.c (K [X1...,Xn]) is irreducible and if in addition K is algebraically closed, then we prove a similar full form of the Hilbert Nullstellensatz for K[X1,...,Xn]. Moreover, if R is a non-field G-domain or R = K [X], where K is an algebraically closed field which is not algebraic over its prime subring, then ∅ ≠ gen (Xi.c(R)) = {S ∈Xi.c(R) | (S : R) = 0}. We determine exactly when the space Xi.c(R) is a Ti - space for i = 0, 1,2. In particular, we show that if Xi:c is T1-space then R is a Hilbert ring and |Xi:c(R)| ≤ 2|Max(R)|. Finally, we determine when the space Xi.c is connected.

LetTbe a countably abelian, totally sub-projective random vari-able. It is well known thatC′′is naturally generic, contra-n-dimensional,meromorphic and contra-Hamilton. We show that there exists a com-pact and differentiablep-adic subring. Unfortunately, we cannot as-sume that every globally connected path is local. It is well known that ̄Pis homeomorphic to Ω.

In this work, we extend an established isomorphism between group rings and a subring of the [Formula: see text] matrices. This extension allows us to construct more complex matrices over the ring [Formula: see text] We present many interesting examples of complex matrices constructed directly from our extension. We also show that some of the matrices used in the literature before can be obtained by a direct application of our extended isomorphism.

The main objective of this paper is to introduce some algebraic properties of finite linear Diophantine fuzzy subsets of group, ring and field. Relatedly, we define the concepts of linear Diophantine fuzzy subgroup and normal subgroup of a group, linear Diophantine fuzzy subring and ideal of a ring, and linear Diophantine fuzzy subfield of a field. We investigate their basic properties, relations and characterizations in detail. Furthermore, we establish the homomorphic images and preimages of the emerged linear Diophantine fuzzy algebraic structures. Finally, we describe linear Diophantine fuzzy code and investigate the relationships between this code and some linear Diophantine fuzzy algebraic structures.

Let R be a commutative ring with identity and $${\mathbb {N}}_0$$ be the additive monoid of nonnegative integers. We say that a function $$t : {\mathbb {N}}_0 \times {\mathbb {N}}_0 \rightarrow R$$ is a twist function on R if t satisfies the following three properties for all $$n, m, q \in {\mathbb {N}}_0$$ : (i) $$t(0,q) = 1$$ , (ii) $$t(n,m) = t(m,n)$$ , and (iii) $$t(n,m) \cdot t(n + m, q) = t (n, m + q) \cdot t(m, q)$$ . Let $$R[\![X]\!]$$ (resp., R[X]) be the set of power series (resp., polynomials) with coefficients in R. For $$f = \sum _{n=0}^{\infty } a_nX^n$$ and $$g = \sum _{n=0}^{\infty } b_nX^n \in R[\![X]\!]$$ , let $$f+g = \sum _{n=0}^{\infty } (a_n+b_n)X^n$$ , $$f*_tg = \sum _{n=0}^{\infty }(\sum _{i+j = n}t(i,j)a_ib_j)X^n$$ . Then, $$R^t[\![X]\!]:= (R[\![X]\!], +, *_t)$$ and $$R^t[X] := (R[X], +, *_t)$$ are commutative rings with identity that contain R as a subring. In this paper, we study ring-theoretic properties of $$R^t[\![X]\!]$$ and $$R^t[X]$$ with focus on divisibility properties including UFDs and GCD-domains. We also show how these two rings are related to the usual power series and polynomial rings.

Let R be a ring such that $$R=R_1+R_2$$ , where $$R_1$$ is a PI subring of R and $$R_2$$ is an additive subgroup of R which satisfies a polynomial identity. We prove that if for some integer $$n\ge 1$$ either $$(R_1R_2)^n \subseteq R_1$$ or $$(R_2R_1)^n \subseteq R_1$$ , then R is a PI ring.

The article takes a look at transcendence and algebraic independence problems, introduces statements and proofs of theorems for some kinds of elements from direct product of 𝑝-adic fields and polynomial estimation theorem. Let Q𝑝 be the 𝑝-adic completion of Q, Ω𝑝 be the completion of the algebraic closure of Q𝑝, 𝑔 = 𝑝1𝑝2 . . . 𝑝𝑛 be a composition of separate prime numbers, Q𝑔 be the 𝑔-adic completion of Q, in other words Q𝑝1 ⊕. . .⊕Q𝑝𝑛. The ring Ω𝑔 ∼=Ω𝑝1⊕...⊕Ω𝑝𝑛, a subring Q𝑔, transcendence and algebraic independence over Q𝑔 are under consideration. Also, hypergeometric series $$𝑓(𝑧) =∞Σ𝑗=0((𝛾1)𝑗 . . . (𝛾𝑟)𝑗)/((𝛽1)𝑗 . . . (𝛽𝑠)𝑗)(𝑧𝑡)^𝑡𝑗 $$, and their formal derivatives are under consideration. Sufficient conditions are obtained under which the values of the series 𝑓(𝛼) and formal derivatives satisfy global relation of algebraic independence, if 𝛼 =∞Σ𝑗=0 𝑎_𝑗𝑔^(𝑟_𝑗), where 𝑎𝑗 ∈ Z𝑔, and non-negative rationals 𝑟𝑗 increase strictly unbounded.

Planar chiral cyclophanopillar[5]arenes with a fused oligo(oxyethylene) or polymethylene subring (MUJs), existing as an equilibrium mixture of subring-included (in) and -excluded (out) conformers, respond to hydrostatic pressure to exhibit dynamic chiroptical property changes, leading to an unprecedented pressure-driven chirality inversion and the largest ever-reported leap of anisotropy (g) factor for the MUJ with a dodecamethylene subring. The pressure susceptivity of MUJs, assessed by the change in g per unit pressure, is a critical function of the size and nature of the subring incorporated and the solvent employed. Mechanistic elucidations reveal that the in–out equilibrium, as the origin of the MUJ's chiroptical property changes, is on a delicate balance of the competitive inclusion of subrings versus solvent molecules as well as the solvation of the excluded subring. The present results further encourage our use of pressure as a unique tool for dynamically manipulating various supramolecular devices/machines., Pressure switches the in/out conformation of cyclophano-pillararenes with accompanying inversion of the chiroptical properties.

В статье рассматриваются вопросы трансцендентности и алгебраической независимости,формулируются и доказываются теоремы длянекоторых элементов прямых произведений $p$-адических полей, а также,теорема об оценке многочлена от таких элементов.Пусть $\mathbb{Q}_p$~--- пополнение $\mathbb{Q}$ по$p$-адической норме, поле $\Omega_{p}$~--- пополнение алгебраического замыкания $\mathbb{Q}_p$,$g=p_1p_2\ldots p_n$~--- произведение различных простых чисел,а пополнение $\mathbb{Q}$ по $g$-адической псевдонормеэто кольцо $\mathbb{Q}_g$, иными словами $\mathbb{Q}_{p_1}\oplus\ldots\oplus\mathbb{Q}_{p_n}$.Рассматривается кольцо $\Omega_g\cong\Omega_{p_1}\oplus\ldots\oplus\Omega_{p_n}$,содержащее $\mathbb{Q}_g$ в качестве подкольца. Вопросы о трансцендентности и алгебраическойнезависимости над $\mathbb{Q}_g$ элементов $\Omega_g$ привели к результатам полученным в статье.При соблюдении некоторых условий можно делать соответствующие выводы для чисел вида$\alpha=\sum\limits_{j=0}^{\infty}a_{j}g^{r_{j}},\;\text{где}\;a_{j}\in \mathbb Z_g,$а неотрицательные рациональные числа $r_{j}$ образуют возрастающую истремящуюся к $+\infty$ при $j\rightarrow +\infty$ последовательность.

We study the structure of nilpotents in relation with a ring property that is near to one-sided duo rings. Such a property is said to be one-sided nilpotent-duo. We prove the following for a one-sided nilpotent-duo ring $R$: (i) The set of nilpotents in $R$ forms a subring; (ii) Köthe's conjecture holds for $R$; (iii) the subring generated by the identity and the set of nilpotents in $R$ is a one-sided duo ring; (iv) if the polynomial ring $R[x]$ over $R$ is one-sided nilpotent-duo then the set of nilpotents in $R$ forms a commutative ring, and $R[x]$ is an NI ring. Several connections between one-sided nilpotent-duo and one-sided duo are given. The structure of one-sided nilpotent-duo rings is also studied in various situations in ring theory. Especially we investigate several kinds of conditions under which one-sided nilpotent-duo rings are NI.

Let R be a commutative ring with identity. We study the concept of pointwise maximal subrings of a ring. A ring R is called a pointwise maximal subring of a ring T if $$R\subset T$$ and for each $$t\in T{\setminus } R$$ , the ring extension $$R[t]\subseteq T$$ has no proper intermediate ring. A characterization of local, integrally closed pointwise maximal subrings of a ring is given. Let G be a subgroup of the group of automorphisms of T. Then the integrally closed pointwise maximality is a G-invariant property of ring extension under some conditions. We also discuss the number of overrings and the Krull dimension of pointwise maximal subrings of a ring. The pointwise maximal subrings of the polynomial ring R[X] are also discussed.

We prove first-order definability of the prime subring inside polynomial rings, whose coefficient rings are (commutative unital) reduced and indecomposable. This is achieved by means of a uniform formula in the language of rings with signature $(0,1,+,\cdot )$. In the characteristic zero case, the claim implies that the full theory is undecidable, for rings of the referred type. This extends a series of results by Raphael Robinson, holding for certain polynomial integral domains, to a more general class.

Let [Formula: see text] be a ring. By the notation [Formula: see text] we denote the additive subgroup of [Formula: see text] generated by all [Formula: see text] in [Formula: see text]. In this work, we partially generalize a result due to Herstein [I. N. Herstein, Topics in Ring Theory (University of Chicago Press, 1969)] showing that if [Formula: see text], then the subring generated by [Formula: see text] is equal to [Formula: see text]. This result implies that [Formula: see text] cannot be a proper subring of [Formula: see text].

Let $$(R,{\mathfrak {m}},\mathbb {k})$$ be an equicharacteristic one-dimensional complete local domain over an algebraically closed field $$\mathbb {k}$$ of characteristic 0. R. Berger conjectured that R is regular if and only if the universally finite module of differentials $$\Omega _R$$ is a torsion-free R module. We give new cases of this conjecture by extending works of Guttes (Arch Math 54:499–510, 1990) and Cortinas et al. (Math Z 228:569–588, 1998). This is obtained by constructing a new subring S of $${\text {Hom}}_R({\mathfrak {m}},{\mathfrak {m}})$$ and constructing enough torsion in $$\Omega _S$$ , enabling us to pull back a nontrivial torsion to $$\Omega _R$$ .

Let R and S be commutative rings with identity, $$f:R\rightarrow S$$ a ring homomorphism and J an ideal of S. Then the subring $$R\bowtie ^fJ:=\{(r,f(r)+j)\mid r\in R$$ and $$j\in J\}$$ of $$R\times S$$ is called the amalgamation of R with S along J with respect to f. In this paper, we generalize and improve recent results on the computation of the diameter of the zero-divisor graph of amalgamated algebras and obtain new results. In particular, we provide new characterizations for completeness of the zero-divisor graph of amalgamated algebra, as well as, a complete description for the diameter of the zero-divisor graph of amalgamations in the special case of finite rings.

A corner of a ring A is a subring eAe, where e is an idempotent. Radical and semi-simple classes which are hereditary for corners and cases where the radical of a ring contains all radical corners are studied.

Associative rings are considered. By a lattice isomorphism, or projection, of a ring R onto a ring Rφ we mean an isomorphism φ of the subring lattice L(R) of R onto the subring lattice L(Rφ) of Rφ. In this case Rφ is called the projective image of a ring R and R is called the projective preimage of a ring Rφ. Let R be a finite ring with identity and Rad R the Jacobson radical of R. A ring R is said to be local if the factor ring R/Rad R is a field. We study lattice isomorphisms of finite local rings. It is proved that the projective image of a finite local ring which is distinct from GF($$ {\mathrm{p}}^{{\mathrm{q}}^{\mathrm{n}}} $$) and has a nonprime residue field is a finite local ring. For the case where both R and Rφ are local rings, we examine interrelationships between the properties of the rings.

Let $\mathcal F_{\mathcal{P}}( L)$ be the set of all frame maps from $\mathcal P(\mathbb R)$ to $L$, which is an $f$-ring. In this paper, we introduce the subrings $\mathcal F_{{\mathcal{P}}_{\infty}}( L)$ of all frame maps from $\mathcal P(\mathbb R)$ to $L$ which vanish at infinity and $\mathcal F_{{\mathcal{P}}_{K}}( L)$ of all frame maps from $\mathcal P(\mathbb R)$ to $L$ with compact support. We prove $\mathcal F_{{\mathcal{P}}_{\infty}}( L)$ is a subring of $\mathcal F_{\mathcal{P}}(L)$ that may not be an ideal of $\mathcal F_{\mathcal{P}}(L)$ in general and we obtain necessary and sufficient conditions for $\mathcal F_{{\mathcal{P}}_{\infty}}( L)$ to be an ideal of $\mathcal F_{\mathcal{P}}( L)$. Also, we show that $\mathcal F_{{\mathcal{P}}_{K}}( L)$ is an ideal of $\mathcal F_{\mathcal{P}}( L)$ and it is a regular ring. For $f\in\mathcal F_{\mathcal{P}}( L)$, we obtain a sufficient condition for $f$ to be an element of $F_{{\mathcal{P}}_{\infty}}( L)$ ($\mathcal F_{{\mathcal{P}}_{K}}( L)$). Next, we give necessary and sufficient conditions for a frame to be compact. We introduce $\mathcal F_{\mathcal{P}}$-pseudocompact and next we establish equivalent condition for an $\mathcal F_{\mathcal{P}}$-pseudocompact frame to be a compact frame. Finally, we study when for some frame $L$ with $\mathcal F_{{\mathcal{P}}_{\infty}} (L)\neq(0)$, there is a locally compact frame $M$ such that $\mathcal F_{{\mathcal{P}}_{\infty}}( L)\cong\mathcal F_{{\mathcal{P}}_{\infty}}(M)$ and $\mathcal F_{{\mathcal{P}}_{K}}( L)\cong\mathcal F_{{\mathcal{P}}_{K}}(M)$.

Let R be a commutative ring with unity. The notion of maximal non valuation domain in an integral domain is introduced and characterized. A proper subring R of an integral domain S is called a maximal non valuation domain in S if R is not a valuation subring of S, and for any ring T such that R ⊂ T ⊂ S, T is a valuation subring of S. For a local domain S, the equivalence of an integrally closed maximal non VD in S and a maximal non local subring of S is established. The relation between dim(R, S) and the number of rings between R and S is given when R is a maximal non VD in S and dim(R, S) is finite. For a maximal non VD R in S such that R ⊂ R'S ⊂ S and dim(R, S) is finite, the equality of dim(R, S) and dim(R'S, S) is established.

Let $$R\subset S$$ be an extension of integral domains. The domain R is said to be a maximal non-integrally closed subring of S if R is not integrally closed in S, while each subring of S properly containing R is integrally closed in S. Jaballah (J Algebra Appl 11(5):1250041, 18pp, 2012) has characterized these domains when S is the quotient field of R. The main purpose of this paper is to study this kind of ring extensions in the general case. Some examples are provided to illustrate our obtained results. Our main result also answers a key question raised by Gilmer and Heinzer (J Math Kyoto Univ 7(2):133–150, 1967).

Let R be a 2-torsion free prime ring with center Z(R) and a nonzero Jordan ideal J. In this paper, our main objective is to prove that: If $$F:R\rightarrow R$$ is a generalized derivation associated with a derivation d of R such that $$[F(u),u]_{k}\in Z(R)$$ for all $$u\in J$$ and a fixed integer $$k\ge 1,$$ then either R is commutative or there exists some $$a\in C$$ such that $$F(x)=ax$$ for all $$x\in R,$$ which extends a result of Soufi and Aboubakr (Turk. J. Math. 38:233–239, 2014, Theorem 3.2). Moreover, a Jordan ideal which is also a subring seems close to an ideal. Therefore, results on Jordan ideals (which are not necessarily subrings) are more interesting. We show that all the theorems of Soufi and Aboubakr [20] are true for Jordan ideals (which are not necessarily subrings).

For a non-zero algebraic integer α, let ℚ(α) denote the simple extension of the field of rational numbers ℚ. ℤ[α] is the smallest subring of ℚ(α) containing both ℤ and α. In this article, we present an account for testing irreducibility of a given polynomial with coefficients in ℤ[α] over the field ℚ(α).

Let $X$ be a nonempty set and ${\mathcal{P}}(X)$ the power set of $X$. The aim of this paper is to identify the unital subrings of ${\mathcal{P}}(X)$ and to compute its cardinality when it is finite. It is proved that any topology $\unicode[STIX]{x1D70F}$ on $X$ such that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$, where $\unicode[STIX]{x1D70F}^{c}=\{U^{c}\mid U\in \unicode[STIX]{x1D70F}\}$, is a unital subring of ${\mathcal{P}}(X)$. It is also shown that $X$ is finite if and only if any unital subring of ${\mathcal{P}}(X)$ is a topology $\unicode[STIX]{x1D70F}$ on $X$ such that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$ if and only if the set of unital subrings of ${\mathcal{P}}(X)$ is finite. As a consequence, if $X$ is finite with cardinality $n\geq 2$, then the number of unital subrings of ${\mathcal{P}}(X)$ is equal to the $n$th Bell number and the supremum of the lengths of chains of unital subalgebras of ${\mathcal{P}}(X)$ is equal to $n-1$.

In this paper, the concept of picture fuzzy subring of a crisp ring is introduced and some related basic results are studied. Also, some properties of picture fuzzy subring under classical ring homomorphism are investigated.

For a saturated fusion system ℱ {\mathcal{F}} on a p-group S, we study the Burnside ring of the fusion system B ⁢ ( ℱ ) {B(\mathcal{F})} , as defined by Matthew Gelvin and Sune Reeh, which is a subring of the Burnside ring B ⁢ ( S ) {B(S)} . We give criteria for an element of B ⁢ ( S ) {B(S)} to be in B ⁢ ( ℱ ) {B(\mathcal{F})} determined by the ℱ {\mathcal{F}} -automorphism groups of essential subgroups of S. When ℱ {\mathcal{F}} is the fusion system induced by a finite group G with S as a Sylow p-group, we show that the restriction of B ⁢ ( G ) {B(G)} to B ⁢ ( S ) {B(S)} has image equal to B ⁢ ( ℱ ) {B(\mathcal{F})} . We also show that, for p = 2 {p=2} , we can gain information about the fusion system by studying the unit group B ⁢ ( ℱ ) × {B(\mathcal{F})^{\times}} . When S is abelian, we completely determine this unit group.

An edge labeled graph is a graph G whose edges are labeled with non-zero ideals of a commutative ring R . A Generalized Spline on an edge labeled graph G is a vertex labeling of G by elements of the ring R , such that the difference between any two adjacent vertex labels belongs to the ideal corresponding to the edge joining both the vertices. The set of generalized splines forms a subring of the product ring R | V | , with respect to the operations of coordinate-wise addition and multiplication. This ring is known as the generalized spline ring R G , defined on the edge labeled graph G , for the commutative ring R . We have considered particular graphs such as complete graphs, complete bipartite graphs and hypercubes, labeling the edges with the non-zero ideals of an integral domain R and have identified the generalized spline ring R G for these graphs. Also, general algorithms have been developed to find these splines for the above mentioned graphs, for any number of vertices and Python code has been written for finding these splines.