Finitely generated mixed modules of Warfield type

In the usual Krull-Schmidt-Azumaya theorem in abelian categories it is essential that each of the direct summands has a local ring of endomorphisms. A partial answer is given here to the case where this last condition is not satisfied by the indecomposable direct summands. It is found that those summands with local endomorphism rings are determined up to isomorphism and cardinality of occurrence.

Let $(R, \mathfrak{m})$ be a Cohen–Macaulay complete local ring. We will apply an inductive argument to show that for every nonprojective locally projective maximal Cohen–Macaulay object $ \mathcal{X} $ of the morphism category of $R$ with local endomorphism ring, there exists an almost split sequence ending in $ \mathcal{X} $. Regular sequences are exploited to reduce the Krull dimension of $R$ on which the inductive argument is established. Moreover, the Auslander–Reiten translate of certain objects is described.

We show that a pointwise finite-dimensional persistence module indexed over a small category decomposes into a direct sum of indecomposables with local endomorphism rings. As an application of this result we give new, short proofs of fundamental structure theorems for persistence modules.

Let P be an R-module with a decomposition P = M 1 ⊕⋯⊕M n , where each M i is a CS-module. In [6], a necessary and sufficient condition for P to be a CS-module for the decomposition is given, by introducing generalized relative injectivity.In this paper, we study the problem when a direct sum of uniform modules is CS for the given direct sum. In the case when each uniform module has a local endomorphism ring, this problem has been studied by several authors, e.g., Baba-Harada [2], Dung [3], Harada-Oshiro [5].

Krull–Schmidt categories and projective covers

For an element $x$ of a commutative ring $R$, let $\Gamma_x(R)$ be the graph whose vertices are the elements of $R$ that divide $x$ such that distinct vertices $r$ and $s$ are adjacent if and only if $rs=x$. If $x$ is a nonzero nonirreducible nonunit of an integral domain $R$, then $\Gamma_x^\mathcal{C}(R)$ is the graph whose vertices are the associate classes of divisors of $x$ that are neither units nor associates of $x$ such that distinct vertices $A$ and $B$ are adjacent if and only if $rs$ divides $x$ for some (and hence every) $r\in A$ and $s\in B$. The graphs $\Gamma_x(R)$ are considered when $R$ is a local Artinian principal ideal ring, and $\Gamma_x^\mathcal{C}(R)$ is examined when $R$ is a valuation domain. For example, it is shown that a finite local ring $R$ is a principal ideal ring if and only if there exists $x\in R$ such that every connected component of $\Gamma_x(R)$ is a star graph of a prescribed cardinality. Moreover, it is proved that an integral domain $R$ is a discrete valuation ring if and only if its collection of graphs $\Gamma_x^\mathcal{C}(R)$ consists precisely of a single-vertex graph, along with every (up to isomorphism) graph that is realizable as the compressed $0$-divisor graph of a local Artinian principal ideal ring. Certain graphs associated with partially ordered abelian groups have an essential role in the work.

The structure of the differential module of a local ring is, of course, much more complicated than that of a field. We wish to determine first its minimal number of generators.

A commutative local ring A of global dimension two conforms to the following description: (a) If the maximal ideal M is either principal or not finitely generated then A is a valuation domain, (b) Otherwise M is generated by a regular sequence of two elements but the ring is not necessarily noetherian.It will be noetherian if and only if it is completely integrally closed.

A study of the application of fast valving and braking resistor for an intelligent SPS

A group in which every element commutes with its endomorphic images is called an “E-group″. If p is a prime number, a p-group G which is an E-group is called a “pE-group″. Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abelian. We conjecture that every finite 3-generator E-group should be abelian. Moreover, we show that the minimum order of a non-abelian pE-group is p 8 for any odd prime number p and this order is 27 for p = 2. Some of these results are proved for a class wider than the class of E-groups.

If I I is an ideal containing a monic polynomial in R [ T ] R[T] where R R is a semilocal ring, then I I and I / I 2 I/{I^2} require the same minimal number of generators. An ideal containing a monic polynomial in a polynomial ring need not possess any minimal set of generators having a monic as a part of it.

Let m ≥ 3 be a positive integer. We prove that there are uncountably many non-commensurable metabelian uniform pro-p groups of dimension m. Consequently, there are uncountably many non-commensurable finitely presented pro-p groups with minimal number of generators m (and minimal number of relations m 2 ${ \binom{m}{2}}$ ).

Suppose G is a nite abelian group with minimal number of generators r .I t is shown that the expected number of elements from G (chosen independently and with the uniform distribution) so that the elements chosen generate G is less than r + ,w here =2 :118456563:::. The constant is explicitly described in terms of the Riemann zeta-function and is best possible.

In this paper we introduce a new class of finitely presented pro-p-groups G of cohomological dimension 2 called mild groups. If d(G), r(G) are respectively the minimal number of generators and relations of G, we give an infinite family of mild groups G with r(G) ≧ d(G) and d(G) ≧ 2 arbitrary. These groups can be constructed with G/[G, G] finite, answering a question of Kuzmin. If G = GS(p) is the Galois group of the maximal p-extension of ℚ unramified outside a finite set of primes S and p ≠ 2, we show that G is mild for a co-final class of sets S, even in the case p ∉ S.

Abstract. Let R be a ring and Q be a finite quiver, and let be the number of vertices of Q. Let be the class of representations of Q by right R-modules with local endomorphism ring and R-module homomorphisms. The endomorphism ring of a representation has at most n maximal right ideals, all of which are also left ideals, and the isomorphism class of M is determined by n invariants. The main theorem of this paper states that a finite direct sum of representations in is unique up to n permutations of m elements. Let . A non-directed graph associated to M is introduced and is shown to determine the unique decomposition of M into indecomposable representations. This class of representations is shown to generalize the known classes of modules for which a theorem analogous to the case of our main theorem holds.