A CHARACTERIZATION OF THE COMMUTATIVE UNITAL RINGS WITH ONLY FINITELY MANY UNITAL SUBRINGS

A (commutative unital) ring is said to have FSP if it has only finitely many unital subrings. The singly generated rings that have FSP have been classified. Thus, a characterization of the rings satisfying FSP is obtained by proving that a ring R has FSP if and only if either R is finite or R = ℤ[t1, …, tn] ⊇ ℤ where ℤ[ti] has FSP for each i = 1, …, n. Also, the following characterization is given for the nontrivial ring direct products Πi ∈ I Ri that have FSP: I is finite, each Ri has FSP, and there is at most one i ∈ I such that Ri has characteristic 0.

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On some orders in ∗-rings based on the core-EP decomposition

Journal of Algebra and Its Applications ◽

10.1142/s0219498822500104 ◽

2020 ◽

pp. 2250010

Author(s):

Janko Marovt ◽

Dijana Mosić

Keyword(s):

Partial Order ◽

The Core ◽

Unital Ring ◽

Rings With Involution ◽

Invertible Elements ◽

Unital Rings ◽

Minus Partial Order

We study certain relations in unital rings with involution that are derived from the core-EP decomposition. The notion of the WG pre-order and the C-E partial order is extended from [Formula: see text], the set of all [Formula: see text] matrices over [Formula: see text], to the set [Formula: see text] of all core-EP invertible elements in an arbitrary unital ring [Formula: see text] with involution. A new partial order is introduced on [Formula: see text] by combining the WG pre-order and the well known minus partial order, and a new characterization of the core-EP pre-order in unital proper ∗-rings is presented. Properties of these relations are investigated and some known results are thus generalized.

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A commutativity theorem for lefts-unital rings

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171290001065 ◽

1990 ◽

Vol 13 (4) ◽

pp. 769-774

Author(s):

Hamza A. S. Abujabal

Keyword(s):

Commutativity Theorem ◽

Unital Ring ◽

Commutativity Theorems ◽

Associative Rings ◽

Unital Rings

In this paper we generalize some well-known commutativity theorems for associative rings as follows: LetRbe a lefts-unital ring. If there exist nonnegative integersm>1,k≥0, andn≥0such that for anyx,yinR,[xky−xnym,x]=0, thenRis commutative.

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On commutativity of one-sideds-unital rings

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171292001078 ◽

1992 ◽

Vol 15 (4) ◽

pp. 813-818

Author(s):

H. A. S. Abujabal ◽

M. A. Khan

Keyword(s):

Polynomial Identity ◽

Unital Ring ◽

Unital Rings

The following theorem is proved: Letr=r(y)>1,s, andtbe non-negative integers. IfRis a lefts-unital ring satisfies the polynomial identity[xy−xsyrxt,x]=0for everyx,y∈R, thenRis commutative. The commutativity of a rights-unital ring satisfying the polynomial identity[xy−yrxt,x]=0for allx,y∈R, is also proved.

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Vertex-Transitive Direct Products of Graphs

The Electronic Journal of Combinatorics ◽

10.37236/6999 ◽

2018 ◽

Vol 25 (2) ◽

Cited By ~ 1

Author(s):

Richard H. Hammack ◽

Wilfried Imrich

Keyword(s):

Direct Product ◽

Direct Products ◽

Odd Cycle ◽

Vertex Transitive ◽

Odd Cycles

It is known that for graphs $A$ and $B$ with odd cycles, the direct product $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. But this is not necessarily true if one of $A$ or $B$ is bipartite, and until now there has been no characterization of such vertex-transitive direct products. We prove that if $A$ and $B$ are both bipartite, or both non-bipartite, then $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. Also, if $A$ has an odd cycle and $B$ is bipartite, then $A\times B$ is vertex-transitive if and only if both $A\times K_2$ and $B$ are vertex-transitive.

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Monomorphic characterization of n-ary direct products

Information Sciences ◽

10.1016/s0020-0255(99)00020-1 ◽

1999 ◽

Vol 119 (3-4) ◽

pp. 275-288 ◽

Cited By ~ 13

Author(s):

J Desharnais

Keyword(s):

Direct Products

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Characterization of ideals of rhotrices over a ring and its applications

Notes on Number Theory and Discrete Mathematics ◽

10.7546/nntdm.2021.27.1.138-147 ◽

2021 ◽

Vol 27 (1) ◽

pp. 138-147

Author(s):

Kailash M. Patil ◽

Keyword(s):

Banach Algebra ◽

Complex Plane ◽

Higher Order ◽

Unital Ring ◽

Maximal Ideals ◽

Unital Banach Algebra

We define higher order rhotrices over a commutative unital ring S and obtain a ring \mathcal{R}_n(S) of rhotrices of the order n \in \mathbb{N}. We characterize the ideals and maximal ideals of \mathcal{R}_n(S). As a particular case, we record ideals of rhotrix rings over integers and rhotrix algebras over complex plane \mathbb{C}. As an application, we characterize the maximal ideals of the commutative unital Banach algebra \mathcal{R}_n(\mathbb{C}).

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On coseparable and 𝔪-coseparable modules

Journal of Algebra and Its Applications ◽

10.1142/s0219498821500316 ◽

2020 ◽

pp. 2150031

Author(s):

Rachid Ech-chaouy ◽

Abdelouahab Idelhadj ◽

Rachid Tribak

Keyword(s):

Direct Summand ◽

Noetherian Rings ◽

Direct Products ◽

Finitely Generated ◽

Direct Sums ◽

Direct Sum ◽

Free Modules

A module [Formula: see text] is called coseparable ([Formula: see text]-coseparable) if for every submodule [Formula: see text] of [Formula: see text] such that [Formula: see text] is finitely generated ([Formula: see text] is simple), there exists a direct summand [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is finitely generated. In this paper, we show that free modules are coseparable. We also investigate whether or not the ([Formula: see text]-)coseparability is stable under taking submodules, factor modules, direct summands, direct sums and direct products. We show that a finite direct sum of coseparable modules is not, in general, coseparable. But the class of [Formula: see text]-coseparable modules is closed under finite direct sums. Moreover, it is shown that the class of coseparable modules over noetherian rings is closed under finite direct sums. A characterization of coseparable modules over noetherian rings is provided. It is also shown that every lifting (H-supplemented) module is coseparable ([Formula: see text]-coseparable).

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Rings in Which the Annihilator of an Ideal Is Pure

Algebra Colloquium ◽

10.1142/s1005386715000796 ◽

2015 ◽

Vol 22 (spec01) ◽

pp. 947-968 ◽

Cited By ~ 5

Author(s):

A. Majidinya ◽

A. Moussavi ◽

K. Paykan

Keyword(s):

Triangular Matrix ◽

Complete Characterization ◽

Direct Products ◽

Matrix Rings ◽

Morita Invariant ◽

Sheaf Representation ◽

Baer Rings ◽

Upper Triangular Matrix ◽

Left Annihilator

A ring R is a left AIP-ring if the left annihilator of any ideal of R is pure as a left ideal. Equivalently, R is a left AIP-ring if R modulo the left annihilator of any ideal is flat. This class of rings includes both right PP-rings and right p.q.-Baer rings (and hence the biregular rings) and is closed under direct products and forming upper triangular matrix rings. It is shown that, unlike the Baer or right PP conditions, the AIP property is inherited by polynomial extensions and has the advantage that it is a Morita invariant property. We also give a complete characterization of a class of AIP-rings which have a sheaf representation. Connections to related classes of rings are investigated and several examples and counterexamples are included to illustrate and delimit the theory.

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