Units in regular abelian p-group rings

In 1963, Gilmer characterized all finite commutative rings with a cyclic group of units and, in 1967, Eldridge and Fischer generalized these results to rings with minimum condition. In the present paper these results are extended to semiperf ect rings and generalizations of the three theorems are obtained. It is shown that a semiperfect ring with cyclic group of units is finite and is either commutative or is the direct sum of a commutative ring and the 2 x 2 upper triangular matrix ring over the field of two elements. Let R be semiperfect with an abelian group of units. It is shown that R is finite if either the group of units is finite or the group of units is finitely generated and the Jacobson radical is nil. The proofs of all these results depend on our main theorem: The structure of a semiperfect ring R with an abelian group of units is described completely up to the structure of commutative local rings. (That is commutative rings with a unique maximal ideal.) The groups of units of these local rings are shown to be direct factors of the group of units of R.

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

Let be a commutative ring with identity. Two elements and b in are called to be associates if and , or equivalently, if . The generalization of associate relation in R has given the idea for definitions of presimplifiable and weakly presimplifiable rings. First of all, it will be given definitions of very strong associate relation, strong regular associate relation, very strongly associate ring, and strongly regular associate ring. The presimplifiable ring is a commutative ring with the condition that every nonzero element is a unit element. While the weakly presimplifiable ring is a commutative ring with the condition that every nonzero element is regular element. Furthermore, it is shown that the relationship between very strongly associate ring with presimplifiable ring and the linkage between strongly regular associate ring and weakly presimplifiable ring. It is obtained that is a presimplifiable ring if and only if is a very strongly associate ring. Meanwhile, is a weakly presimplifiable ring if and only if is a strongly regular associate ring. Then, it is shown that the correlation between presimplifiable and weakly presimplifiable rings to its polynomial ring and its the formal power series ring . If is a weakly presimplifiable ring, then and are also weakly presimplifiable rings. However, if is a presimplifiable ring, then is also a presimplifiable ring but always not valid for .

In this paper some conditions for a skew group ring or a crossed product to have finite weak global dimension are given.Using these results we obtain some necessary conditions and some sufficient conditions for a skew group ring or a crossed product to be a Dubrovin valuation ring.If R*G is a skew group ring, where the coefficient ring R is a commutative ring and G is a finite group, then we prove that the conditions we obtained become necessary and sufficient conditions.In particular, if R is a commutative valuation ring, then R*G is a Dubrovin valuation ring if and only if G T=<1>,where G T is the inertial group of R.

A commutative Noetherian local ring of finite global dimension is a regular local ring and the structure of such a ring is well described 1n many texts (see §1.9). It is perhaps surprising that very little seems to be known about the corresponding non-commutat1ve rings and it is our aim 1n this thesis to examine some of the ways in which the commutative theory can, and cannot be extended to a right Noetherian local ring of finite right global dimension. Chapter 1 contains basic definitions and, in Chapter 2, results are obtained on the projective and Injective dimensions of modules over right Noetherian local rings. Commutative regular local rings are domains and we begin Chapter 3 by considering the question of when a right Noetherian local ring R of finite right global dimension is a prime ring. An example of Stafford's shows that R is not necessarily prime; however, by examining the Nilpotent radical of R, we are able to extend results of Ramras [57] and Walker [75] and show that R is indeed prime when certain prime ideals are localisable. In Chapter 4 we consider the lattice of prime ideals of R when R is an AR-r1ng and provide theorems which generalise some of the basic results from the theory of commutative regular local rings. Section 4.3 contains examples which not only illustrate points arising in Chapters 3 and 4 but also show that some of the techniques which are mainstays of the commutative theory, fall dramatically in a non-commutat1ve setting. In Chapter 5, we generalise the concepts of regular sequences and Cohen Macaulay rings enabling us to prove, in §5.2 and chapter 6, that a right Noetherian local ring of finite right global dimension, which 1s integral over its centre, is a prime ring and exhibits many properties similar to those enjoyed by commutative regular local rings. Examples are provided which show that some alternative approaches are not applicable to the rings considered in these chapters. Regular local rings are Gorensteln rings and, as such, are the subject of an elegant structure theorem due to Bass [5]. In Chapter 7, we generalise his theorem to the situation of rings Integral over their centres. Each chapter begins with a summary of the results contained in that chapter.

In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. The method of rigidity was modified to work over general commutative base rings in our paper (Yekutieli and Zhang, Trans AMS 360:3211-3248, 2008). In the present paper we obtain many of the important local features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results apply to essentially finite type algebras over a regular noetherian finite dimensional base ring, and hence are suitable for arithmetic rings. In the sequel paper (Yekutieli, Rigid dualizing complexes on schemes, in preparation) these results will be used to construct and study rigid dualizing complexes on schemes.

Let R be a ring with identity of characteristic two and G a nontrivial torsion group. We show that if the units in the group ring $RG$ are all trivial, then G must be cyclic of order two or three. We also consider the case where R is a commutative ring with identity of odd prime characteristic and G is a nontrivial locally finite group. We show that in this case, if the units in $RG$ are all trivial, then G must be cyclic of order two. These results improve on a result of Herman et al. [‘Trivial units for group rings with G-adapted coefficient rings’, Canad. Math. Bull.48(1) (2005), 80–89].

Cartan maps, clean rings, and unique factorization

A commutative ring R is called J-rad clean in case, for any r ∈ R, there is an idempotent e ∈ R such that r−e ∈ U(R) and re ∈ J(R), where U(R) and J(R) denote the set of units and the Jacobson radical of R, respectively. Also, a ring R is called semiregular if R/J(R) is regular in the sense of von Neumann and idempotents lift modulo J(R). We demonstrate that these two concepts are, actually, equivalent and we portray a portion of the properties of this class of rings. In particular, as a direct application, we prove that the commutative group ring RG is J-rad clean if, and only if, R is a commutative J-rad clean ring and G is a torsion abelian group, provided that J(R) is nil.

Endoregular modules

Let G be an abelian group and let R be a commutative ring with identity. Denote by R t G a commutative twisted group algebra (a commutative twisted group ring) of G over R, by ℬ(R) and ℬ(R t G) the nil radicals of R and R t G, respectively, by G p the p-component of G and by G 0 the torsion subgroup of G. We prove that: i. If R is a ring of prime characteristic p, the multiplicative group R* of R is p-divisible and ℬ(R) = 0, then there exists a twisted group algebra R t 1 (G/G p ) such that R t G/ℬ(R t G) ≅ R t 1 (G/G p ) as R-algebras; ii. If R is a ring of prime characterisitic p and R* is p-divisible, then ℬ(R t G) = 0 if and only if ℬ(R) = 0 and G p = 1; and iii. If B(R) = 0, the orders of the elements of G 0 are not zero divisors in R, H is any group and the commutative twisted group algebra R t G is isomorphic as R-algebra to some twisted group algebra R t 1 H, then R t G 0 ≅ R t 1 H 0 as R-algebras.

<span>The notion of the matrix Jacobson graph was introduced in 2019. Let </span><span class="math inline"><em>R</em></span><span> be a commutative ring and </span><span class="math inline"><em>J</em>(<em>R</em>)</span><span> be the Jacobson radical of ring </span><span class="math inline"><em>R</em></span><span>. The matrix Jacobson graph of ring </span><span class="math inline"><em>R</em></span><span> size </span><span class="math inline"><em>m</em> × <em>n</em></span><span>, denoted </span><span class="math inline">𝔍(<em>R</em>)^{<em>m</em> × <em>n</em>}</span><span>, is defined as a graph where the vertex set is </span><span class="math inline"><em>R</em>^{<em>m</em> × <em>n</em>} ∖ <em>J</em>(<em>R</em>)^{<em>m</em> × <em>n</em>}</span><span> such that two distinct vertices </span><span class="math inline"><em>A</em>, <em>B</em></span><span> are adjacent if and only if </span><span class="math inline">1 − det(<em>A</em>^<em>t</em><em>B</em>)</span><span> is not a unit in ring </span><span class="math inline"><em>R</em></span><span>. Here we obtain some graph theoretical properties of </span><span class="math inline">𝔍(<em>R</em>)^{<em>m</em> × <em>n</em>}</span><span> including its connectivity, planarity and perfectness.</span>

Let \Lambda be a finite abelian group. A dynamical system with transformation group \Lambda is a triple (A,\Lambda,\alpha) , consisting of a unital locally convex algebra A , the finite abelian group \Lambda and a group homomorphism \alpha:\Lambda\rightarrow\operatorname{Aut}(A) , which induces an action of \Lambda on A . In this paper we present a new, geometrically oriented approach to the noncommutative geometry of principal bundles with finite abelian structure group based on such dynamical systems.

Let G be an abelian group, R a commutative ring of prime characteristic p with identity and R t G a commutative twisted group ring of G over R. Suppose p is a fixed prime, G p and S(R t G) are the p-components of G and of the unit group U(R t G) of R t G, respectively. Let R* be the multiplicative group of R and let f α(S) be the α-th Ulm-Kaplansky invariant of S(R t G) where α is any ordinal. In the paper the invariants f n (S), n ∈ ℕ ∪ {0}, are calculated, provided G p = 1. Further, a commutative ring R with identity of prime characteristic p is said to be multiplicatively p-perfect if (R*)p = R*. For these rings the invariants f α(S) are calculated for any ordinal α and a description, up to an isomorphism, of the maximal divisible subgroup of S(R t G) is given.

We calculate, only in terms of a commutative unital ring R of prime characteristic p and an abelian p-mixed group G, the classical Warfield q-invariants Wα,q(VR(G)) of the group VR(G) of all normalized units in the group ring R(G). This continues our results in (Extr. Math., 2005), (Collect. Math., 2008) and (J. Alg. Appl., 2008).