Commutative Local Ring Research Articles

EXTENSIONS OF HOMOGENEOUS SEMILOCAL RINGS I

A ring R with Jacobson radical J(R) is a homogeneous semilocal ring if R/J(R) is simple artinian. In this paper, we study the transfer of the property of being homogeneous semilocal from a ring R to the formal power series ring R[[x]], the skew formal power series ring R[[x, α]] and the Hurwitz series ring HR. The results of the paper generalize those proved for commutative local rings. We also consider finite centralizing extensions proving that if the ring of matrices Mn(R) is a homogeneous semilocal ring, then so is R. More generally, if e is an idempotent of a homogeneous semilocal ring S, then eSe is homogeneous semilocal.

Nil-quasipolar rings

Let \(R\) be an arbitrary ring. An element \(a\in R\) is nil-quasipolar if there exists \(p^2=p\in comm^2(a)\) such that \(a+p\in Nil(R)\); \(R\) is called nil-quasipolar in case each of its elements is nil-quasipolar. In this paper, we study nil-quasipolar rings over commutative local rings. We determine the conditions under which a single \(2\times 2\) matrix over a commutative local ring is nil-quasipolar. It is shown that \(A\in M_2(R)\) is nil-quasipolar if and only if \(A\in Nil\big (M_2(R)\big )\) or \(A+I_2\in Nil\big (M_2(R)\big )\) or the characteristic polynomial \(\chi (A)\) has a root in \(Nil(R)\) and a root in \(-1+Nil(R)\). We give some equivalent characterizations of nil-quasipolar rings through the endomorphism ring of a module. Among others we prove that every nil-quasipolar ring has stable range one.

Automorphisms of Stable Linear Groups Over Commutative Local Rings With 1/2

In the paper, we describe automorphisms of stable linear groups E(R) and GL(R) over commutative local rings R with 1/2.

Automorphisms of Chevalley Groups of Type G 2 Over Local Rings Without 1/2

In this paper, we prove that every automorphism of a Chevalley group of type G 2 over a commutative local ring without 1/2 is the composition of a ring automorphism and a conjugation by some matrix.

Vanishing of Tate Cohomology and Gorenstein Injective Dimension

In this note, some sufficient conditions are given for the equality Gid R RHom R (M, N) = G-dim R M + Gid R N to hold via vanishing of Tate cohomology. For instance, let R be a commutative Gorenstein local ring, and let M and N be finite R-modules. If one has for all i ∈ ℤ, then Gid R RHom R (M, N) = G-dim R M + Gid R N. In addition, some Auslander–Buchsbaum type width (depth) formulas are given.

The negative side of cohomology for Calabi-Yau categories

We study Z-graded cohomology rings defined over Calabi–Yau categories. We show that the cohomology in negative degree is a trivial extension of the cohomology ring in non-negative degree, provided the latter admits a regular sequence of central elements of length 2. In particular, the products of elements of negative degrees are zero. As corollaries, we apply this to Tate–Hochschild cohomology rings of symmetric algebras, and to Tate cohomology rings over group algebras. We also prove similar results for Tate cohomology rings over commutative local Gorenstein rings.

Strongly clean matrices over arbitrary rings

We characterize when the companion matrix of a monic polynomial over an arbitrary ring R is strongly clean, in terms of a type of ideal-theoretic factorization (which we call an iSRC factorization) in the polynomial ring R[t]. This provides a nontrivial necessary condition for Mn(R) to be strongly clean, for R arbitrary. If the ring in question is either local or commutative, then we can say more (generalizing and extending most of what is currently known about this problem). If R is local, our iSRC factorization is equivalent to an actual polynomial factorization, generalizing results in [1], [18] and [12]. If, instead, R is commutative and h∈R[t] is monic, we again show that an iSRC factorization yields a polynomial factorization, and we prove that h has such a factorization if and only if its companion matrix is strongly clean, if and only if every algebraic element (in every R-algebra) which satisfies h is strongly clean. This generalizes the work done in [1] on commutative local rings and provides a characterization of strong cleanness in Mn(R) for any commutative ring R.

SOME RESULTS ON COZERO-DIVISOR GRAPH OF A COMMUTATIVE RING

Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ′(R) is a graph with the vertex set W*(R), where W*(R) is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a ∉ Rb and b ∉ Ra. In this paper, we show that if Γ′(R) is a forest, then Γ′(R) is a union of isolated vertices or a star. Also, we prove that if Γ′(R) is a forest with at least one edge, then R ≅ ℤ2 ⊕ F, where F is a field. Among other results, it is shown that for every commutative ring R, diam (Γ′(R[x])) = 2. We prove that if R is a field, then Γ′(R[[x]]) is totally disconnected. Also, we prove that if (R, m) is a commutative local ring and m ≠ 0, then diam (Γ′(R[[x]])) ≤ 3. Finally, it is proved that if R is a commutative non-local ring, then diam (Γ′(R[[x]])) ≤ 3.

A Note on Invariants Over Finite Local Rings

In this note, we prove a conjecture of Steinberg [10] about modular invariants of some linear groups over finite commutative local rings.

Commutative Local Rings Whose Ideals are Direct Sums of Cyclic Modules

A well-known result of Köthe and Cohen-Kaplansky states that a commutative ring R has the property that every R-module is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring. This motivated us to study commutative rings for which every ideal is a direct sum of cyclic modules. Recently, in Behboodi et al. Commutative Noetherian local rings whose ideals are direct sums of cyclic modules (J. Algebra 345:257–265, 2011) the authors considered this question in the context of finite direct products of commutative Noetherian local rings. In this paper, we continue their study by dropping the Noetherian condition.

Inverse problems for deformation rings

Let W W be a complete Noetherian local commutative ring with residue field k k of positive characteristic p p . We study the inverse problem for the universal deformation rings R W ( Γ , V ) R_{W}(\Gamma ,V) relative to W W of finite dimensional representations V V of a profinite group Γ \Gamma over k k . We show that for all p p and n ≥ 1 n \ge 1 , the ring W [ [ t ] ] / ( p n t , t 2 ) W[[t]]/(p^n t,t^2) arises as a universal deformation ring. This ring is not a complete intersection if p n W ≠ { 0 } p^nW\neq \{0\} , so we obtain an answer to a question of M. Flach in all characteristics. We also study the ‘inverse inverse problem’ for the ring W [ [ t ] ] / ( p n t , t 2 ) W[[t]]/(p^n t,t^2) ; this is to determine all pairs ( Γ , V ) (\Gamma , V) such that R W ( Γ , V ) R_{W}(\Gamma ,V) is isomorphic to this ring.

The Zero Divisor Graphs of Commutative Local Rings of Order p 4 and p 5

To each commutative ring R we can associate a zero divisor graph whose vertices are the zero divisors of R and such that two vertices are adjacent if their product is zero. Detecting isomorphisms among zero divisor graphs can be reduced to the problem of computing the classes of R under a suitable semigroup congruence. Presently, we introduce a strategy for computing this quotient for local rings using knowledge about a generating set for the maximal ideal. As an example, we then compute Γ(R) for several classes of rings; with the results in [4] these classes include all local rings of order p 4 and p 5 for prime p.

Nekovár duality over p-adic Lie extensions of global fields

Tate duality is a Pontryagin duality between the i th Galois cohomology group of the absolute Galois group of a local field with coefficents in a finite module and the (2-i) th cohomology group of the Tate twist of the Pontryagin dual of the module. Poitou-Tate duality has a similar formulation, but the duality now takes place between Galois cohomology groups of a global field with restricted ramification and compactly-supported cohomology groups. Nekovár proved analogues of these in which the module in question is a finitely generated module T over a complete commutative local Noetherian ring R with a commuting Galois action, or a bounded complex thereof, and the Pontryagin dual is replaced with the Grothendieck dual T^* , which is a bounded complex of the same form. The cochain complexes computing the Galois cohomology groups of T and T^*(1) are then Grothendieck dual to each other in the derived category of finitely generated R -modules. Given a p -adic Lie extension of the ground field, we extend these to dualities between Galois cochain complexes of induced modules of T and T^*(1) in the derived category of finitely generated modules over the possibly noncommutative Iwasawa algebra with R -coefficients.

STRONGLY CLEAN MATRICES OVER COMMUTATIVE LOCAL RINGS

An element of a ring is called strongly clean provided that it can be written as the sum of an idempotent and a unit that commute. We characterize, in this paper, the strongly cleanness of matrices over commutative local rings. This partially extend many known results such as Theorem 12 in Borooah, Diesl and Dorsey [Strongly clean matrix rings over commutative local rings, J. Pure Appl. Algebra212 (2008) 281–296], Theorem 3.2.7 and Proposition 3.3.6 in Dorsey [Cleanness and strong cleanness of rings of matrices, Ph.D. thesis, University of California, Berkeley (2006)], Theorem 2.3.14 in Fan [Algebraic analysis of some strongly clean and their generalization, Ph.D. thesis, Memorial University of Newfoundland, Newfoundland (2009)], Theorem 3.1.9 and Theorem 3.1.26 in Yang [Strongly clean rings and g(x)-clean rings, Ph.D. thesis, Memorial University of Newfoundland, Newfoundland (2007)].

On Quasipolar Rings

The notion of quasipolar elements of rings was introduced by Koliha and Patricio in 2002. In this paper, we introduce the notion of quasipolar rings and relate it to other familiar notions in ring theory. It is proved that both strongly π-regular rings and uniquely clean rings are quasipolar, and quasipolar rings are strongly clean, but no two of these classes of rings are equivalent. For commutative rings, quasipolar rings coincide with semiregular rings. It is also proved that every n × n upper triangular matrix ring over any commutative uniquely clean ring or commutative local ring is quasipolar.

Construction and decoding of BCH codes over chain of commutative rings

Abstract In this paper, we present a new construction and decoding of BCH codes over certain rings. Thus, for a nonnegative integer t, let A 0 ⊂ A 1 ⊂ ⋯ ⊂ A t − 1 ⊂ A t be a chain of unitary commutative rings, where each A i is constructed by the direct product of appropriate Galois rings, and its projection to the fields is K 0 ⊂ K 1 ⊂ ⋯ ⊂ K t − 1 ⊂ K t (another chain of unitary commutative rings), where each K i is made by the direct product of corresponding residue fields of given Galois rings. Also, A i ∗ and K i ∗ are the groups of units of A i and K i , respectively. This correspondence presents a construction technique of generator polynomials of the sequence of Bose, Chaudhuri, and Hocquenghem (BCH) codes possessing entries from A i ∗ and K i ∗ for each i, where 0 ≤ i ≤ t. By the construction of BCH codes, we are confined to get the best code rate and error correction capability; however, the proposed contribution offers a choice to opt a worthy BCH code concerning code rate and error correction capability. In the second phase, we extend the modified Berlekamp-Massey algorithm for the above chains of unitary commutative local rings in such a way that the error will be corrected of the sequences of codewords from the sequences of BCH codes at once. This process is not much different than the original one, but it deals a sequence of codewords from the sequence of codes over the chain of Galois rings.

A Class of Quasipolar Rings

An element a of a ring R is called J-quasipolar if there exists p 2 = p ∈ R satisfying p ∈ comm2(a) and a + p ∈ J(R); R is called J-quasipolar in case each of its elements is J-quasipolar. The class of this sort of rings lies properly between the class of uniquely clean rings and the class of quasipolar rings. In particular, every J-quasipolar element in a ring is quasipolar. It is shown, in this paper, that a ring R is J-quasipolar iff R/J(R) is boolean and R is quasipolar. For a local ring R, we prove that every n × n upper triangular matrix ring over R is J-quasipolar iff R is uniquely bleached and R/J(R) ≅ ℤ2. Moreover, it is proved that any matrix ring of size greater than 1 is never J-quasipolar. Consequently, we determine when a 2 × 2 matrix over a commutative local ring is J-quasipolar. A criterion in terms of solvability of the characteristic equation is obtained for such a matrix to be J-quasipolar.

On the Krull Dimensions of the Steenrod Algebra

For any prime p, we show that the mod p Steenrod algebra (a local ring with nil maximal ideal) has infinite little Krull dimension. This contrasts sharply with the case of a commutative (or noetherian) local ring with nil maximal ideal which must have little Krull dimension equal to 0. Also, we show that the Steenrod algebra has no Krull dimension, classical Krull dimension, or Gabriel dimension.

Are complete intersections complete intersections?

A commutative local ring is generally defined to be a complete intersection if its completion is isomorphic to the quotient of a regular local ring by an ideal generated by a regular sequence. It has not previously been determined whether or not such a ring is necessarily itself the quotient of a regular ring by an ideal generated by a regular sequence. In this article, it is shown that if a complete intersection is a one dimensional integral domain, then it is such a quotient. However, an example is produced of a three dimensional complete intersection domain which is not a homomorphic image of a regular local ring, and so the property does not hold in general.

Strong cleanness of generalized matrix rings over a local ring

Let Ks(R) be the generalized matrix ring over a ring R with multiplier s. For a general local ring R and a central element s in the Jacobson radical of R, necessary and sufficient conditions are obtained for Ks(R) to be a strongly clean ring. For a commutative local ring R and an arbitrary element s in R, criteria are obtained for a single element of Ks(R) to be strongly clean and, respectively, for the ring Ks(R) to be strongly clean. Specializing to s=1 yields some known results. New families of strongly clean rings are presented.