On Reduced Modules

Proceedings of The Japan Academy Series A-mathematical Sciences, 1987

Introduction. Let R be a fixed (not necessarily commutative) ring.

Journal of Mathematics of Kyoto University

Progress in Mathematics, 1998

Journal of Algebra and Its Applications, 2012

For a ring derivation δ, we introduce and investigate a generalization of reduced rings and Armendariz rings which we call a δ-Armendariz ring. Various classes of δ-Armendariz rings is provided and a number of properties of this generalization are established. Radicals and minimal prime ideals of the differential polynomial ring R[x; δ], in terms of those of a δ-Armendariz R, is determined. We prove that several properties transfer between R and the differential polynomial ring R[x; δ], in case R is δ-Armendariz.

Hacettepe Journal of Mathematics and Statistics, 2015

Let R be a ring with identity and J(R) denote the Jacobson radical of R. In this paper, we introduce a new class of rings called feckly reduced rings. The ring R is called feckly reduced if R/J(R) is a reduced ring. We investigate relations between feckly reduced rings and other classes of rings. We obtain some characterizations of being a feckly reduced ring. It is proved that a ring R is feckly reduced if and only if every cyclic projective R-module has a feckly reduced endomorphism ring. Among others we show that every left Artinian ring is feckly reduced if and only if it is 2-primal, R is feckly reduced if and only if T (R, R) is feckly reduced if and only if R[x]/ < x 2 > is feckly reduced.

Hacettepe Journal of Mathematics and Statistics, 2014

In this paper, we study the behavior of endomorphism rings of a cyclic, finitely presented module of projective dimension ≤ 1. This class of modules extends to arbitrary rings the class of couniformly presented modules over local rings.

Formalized Mathematics

Summary We formalize in the Mizar system [3], [4] some basic properties on left module over a ring such as constructing a module via a ring of endomorphism of an abelian group and the set of all homomorphisms of modules form a module [1] along with Ch. 2 set. 1 of [2]. The formalized items are shown in the below list with notations: Mab for an Abelian group with a suffix “ ab ” and M without a suffix is used for left modules over a ring. 1. The endomorphism ring of an abelian group denoted by End(Mab ). 2. A pair of an Abelian group Mab and a ring homomorphism R → ρ R\mathop \to \limits^\rho End (Mab ) determines a left R-module, formalized as a function AbGrLMod(Mab, ρ) in the article. 3. The set of all functions from M to N form R-module and denoted by Func_Mod R (M, N). 4. The set R-module homomorphisms of M to N, denoted by Hom R (M, N), forms R-module. 5. A formal proof of Hom R (¯R, M) ≅M is given, where the ¯R denotes the regular representation of R, i.e. we regard R itself a...

Journal of The Korean Mathematical Society, 2019

Let R be a commutative ring with unity. If f (x) is a zerodivisor polynomial such that f (x) = c f f 1 (x) with c f ∈ R and f 1 (x) is not zero-divisor, then c f is called an annihilating content for f (x). In this case Ann(f) = Ann(c f). We defined EM-rings to be rings with every zero-divisor polynomial having annihilating content. We showed that the class of EM-rings includes integral domains, principal ideal rings, and PP-rings, while it is included in Armendariz rings, and rings having a.c. condition. Some properties of EM-rings are studied and the zerodivisor graphs Γ(R) and Γ(R[x]) are related if R was an EM-ring. Some properties of annihilating contents for polynomials are extended to formal power series rings.

International Electronic Journal of Algebra

This is the first in a series of papers highlighting the applications of reduced and coreduced modules. Let $R$ be a commutative unital ring and $I$ be an ideal of $R$. We show that $I$-reduced $R$-modules and $I$-coreduced $R$-modules provide a setting in which the Matlis-Greenless-May (MGM) Equivalence and the Greenless-May (GM) Duality hold. These two notions have been hitherto only known to exist in the derived category setting. We realise the $I$-torsion and the $I$-adic completion functors as representable functors and under suitable conditions compute natural transformations between them and other functors.

Al-Jabar : Jurnal Pendidikan Matematika

Let R commutative ring with multiplicative identity, and M is an R-module. An ideal I of R is irreducible if the intersection of every two ideals of R equals I, then one of them is I itself. Module theory is also known as an irreducible submodule, from the concept of an irreducible ideal in the ring. The set of R - module homomorphisms from M to itself is denoted by EndR(M). It is called a module endomorphism M of ring R. The set EndR(M) is also a ring with an addition operation and composition function. This paper showed the sufficient condition of an irreducible ideal on the ring of EndR(R) and EndR(M)

Bulletin of the American Mathematical Society, 1973

Let R be a commutative ring. A finitely generated R-module M can be converted into an R(X)−module by an R−endomorphism of M (see for example (4)). In this work, we first give a structure Theorem for finitely generated modules over local rings in term of Fitting ideals. And then we consider an R(X)/(f(X))−finitely generated module Mu,f induced on M by an endomorphism u which annihilate a monic poly- nomial f(X). We establish a structure Theorem for Mu,f which shall have interesting applications in linear algebra.

Journal of Mathematical Sciences, 2009

All right R-modules are I0-modules if and only if either R is a right SV-ring or R/I (2) (R) is an Artinian serial ring such that the square of the Jacobson radical of R/I (2) (R) is equal to zero. All rings are assumed to be associative and with nonzero identity element. Expressions such as "an Artinian ring" mean that the corresponding right and left conditions hold. A submodule X of the module M is said to be superfluous in M if X + Y = M for every proper submodule P of the module M. Following [9], we call a module M an I 0-module if every nonsuperfluous submodule of M contains a nonzero direct summand of the module M. It is clear that I 0-modules are weakly regular modules, considered in [1-3, 8]; a module M is said to be weakly regular if every submodule of M that is not contained in the Jacobson radical of M contains a nonzero direct summand of M. Weakly regular modules are studied in [1-3; 6; 8; 9; 11, Chap. 3; 12-14], and other papers. A ring R is called a right generalized SV-ring if every right R-module is weakly regular. It is clear that A is a right generalized SV-ring provided each right A-module is an I 0-module. In addition, it follows from the presented paper that the Jacobson radical of any right module over a right generalized SV-ring is superfluous; therefore, every right module over a right generalized SV-ring is an I 0-module. The aim of the paper is the study of generalized right SV-rings. The main result of the present paper is Theorem 1.

Journal of Algebra, 1979

We prove here, among other results, that if R is a commutative noetherian ring and proJective R[xx, ,..., x,]-modules of rank < Krull dim R are extended, then finitely generated projective R[x, ,..., x,]-modules are extended. We also give an example of a nonfinitely generated projective module over an integral domain which contains no unimodular elements. All the rings in this paper are associative and commutative with unit and the algebras are associative with unit. We present first some variants of Quillen's theorem [lo, Theorem 11. Our arguments in this part are based on a solution of Serre's problem, communicated to the author by Professor L.N. Vasergtein (Moscow) and on the proof of [lo, Theorem 11. If S is a multiplicative set in a ring R, M and R-module and m EM, we denote by m, the image of m in M, under the canonical homomorphism M + MS. As usual if m is a maximal ideal of R we use the notation

Journal of the Australian Mathematical Society, 2017

Our main purpose is to extend several results of interest that have been proved for modules over integral domains to modules over arbitrary commutative rings $R$ with identity. The classical ring of quotients $Q$ of $R$ will play the role of the field of quotients when zero-divisors are present. After discussing torsion-freeness and divisibility (Sections 2–3), we study Matlis-cotorsion modules and their roles in two category equivalences (Sections 4–5). These equivalences are established via the same functors as in the domain case, but instead of injective direct sums $\oplus Q$ one has to take the full subcategory of $Q$-modules into consideration. Finally, we prove results on Matlis rings, i.e. on rings for which $Q$ has projective dimension $1$ (Theorem 6.4).