On the associativity of the torsion functor

1974 ◽  
Vol 10 (1) ◽  
pp. 107-118
Author(s):  
John Clark
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Noetherian Ring ◽  
Integral Domain ◽  
Direct Sum ◽  
Torsion Functor

Let R be a commutative ring with identity. We say that tor is associative over R if for all R-modules A, B, C there is an isomorphism Our main results are that (1) tor is associative over a noetherian ring R if and only if R is the direct sum of a finite number of Dedekind rings and uniserial rings, and (2) tor is associative over an integral domain R if and only if R is a Prüfer ring.

Primary Ideals and Prime Power Ideals

1966 ◽  
Vol 18 ◽  
pp. 1183-1195 ◽  
Author(s):  
H. S. Butts ◽  
Robert W. Gilmer
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Maximal Ideal ◽  
Integral Domain ◽  
Prime Power ◽  
Ideal Theory ◽  
Dedekind Domain ◽  
Primary Ideal ◽  
Primary Ideals ◽  
The Ideal

This paper is concerned with the ideal theory of a commutative ringR.We sayRhas Property (α) if each primary ideal inRis a power of its (prime) radical;Ris said to have Property (δ) provided every ideal inRis an intersection of a finite number of prime power ideals. In (2, Theorem 8, p. 33) it is shown that ifDis a Noetherian integral domain with identity and if there are no ideals properly between any maximal ideal and its square, thenDis a Dedekind domain. It follows from this that ifDhas Property (α) and is Noetherian (in which caseDhas Property (δ)), thenDis Dedekind.

scholarly journals THE ANNIHILATING-IDEAL GRAPH OF COMMUTATIVE RINGS II

2011 ◽  
Vol 10 (04) ◽  
pp. 741-753 ◽  
Author(s):  
M. BEHBOODI ◽  
Z. RAKEEI
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Noetherian Ring ◽  
Undirected Graph ◽  
Commutative Rings ◽  
Complete Characterization ◽  
Zero Divisors ◽  
Reduced Ring

In this paper we continue our study of annihilating-ideal graph of commutative rings, that was introduced in (The annihilating-ideal graph of commutative rings I, to appear in J. Algebra Appl.). Let R be a commutative ring with 𝔸(R) be its set of ideals with nonzero annihilator and Z(R) its set of zero divisors. The annihilating-ideal graph of R is defined as the (undirected) graph 𝔸𝔾(R) that its vertices are 𝔸(R)* = 𝔸(R)\{(0)} in which for every distinct vertices I and J, I — J is an edge if and only if IJ = (0). First, we study the diameter of 𝔸𝔾(R). A complete characterization for the possible diameter is given exclusively in terms of the ideals of R when either R is a Noetherian ring or Z(R) is not an ideal of R. Next, we study coloring of annihilating-ideal graphs. Among other results, we characterize when either χ(𝔸𝔾(R)) ≤ 2 or R is reduced and χ(𝔸𝔾(R)) ≤ ∞. Also it is shown that for each reduced ring R, χ(𝔸𝔾(R)) = cl (𝔸𝔾(R)). Moreover, if χ(𝔸𝔾(R)) is finite, then R has a finite number of minimal primes, and if n is this number, then χ(𝔸𝔾(R)) = cl (𝔸𝔾(R)) = n. Finally, we show that for a Noetherian ring R, cl (𝔸𝔾(R)) is finite if and only if for every ideal I of R with I2 = (0), I has finite number of R-submodules.

scholarly journals Primitiewe elemente vir kommutatiewe ringuitbreidings

1991 ◽  
Vol 10 (2) ◽  
pp. 67-71
Author(s):  
H. J. Schutte
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Integral Domain ◽  
Prime Ideals ◽  
Primitive Elements ◽  
Integral Domains ◽  
Field Extensions ◽  
Ring Extensions ◽  
Minimal Prime

The existence of primitive elements for integral domain extensions is considered with reference to the well known theorem about primitive elements for field extensions. Primitive elements for extensions of a commutative ring R with identity are considered, where R has only a finite number of minimal prime ideals with zero intersection. This case is reduced to the case for ring extensions of integral domains.

scholarly journals On a class of dual Rickart modules

2020 ◽  
Vol 72 (7) ◽  
pp. 960-970
Author(s):  
R. Tribak
Keyword(s):  
Direct Summand ◽  
Noetherian Ring ◽  
Commutative Noetherian Ring ◽  
Direct Sum ◽  
Finite Set ◽  
Rickart Modules

UDC 512.5 Let R be a ring and let Ω R be the set of maximal right ideals of R . An R -module M is called an sd-Rickart module if for every nonzero endomorphism f of M , ℑ f is a fully invariant direct summand of M . We obtain a characterization for an arbitrary direct sum of sd-Rickart modules to be sd-Rickart. We also obtain a decomposition of an sd-Rickart R -module M , provided R is a commutative noetherian ring and A s s ( M ) ∩ Ω R is a finite set. In addition, we introduce and study ageneralization of sd-Rickart modules.

Field, commutative ring, integral domain

2021 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Knowledge Base ◽  
Commutative Ring ◽  
Integral Domain ◽  
White Paper

FIELD, three propositions, and their underlying definitions are presented in this white paper (knowledge base).

RINGS: Almost a ring, semiring, zero, integral domain

2021 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Knowledge Base ◽  
Commutative Ring ◽  
Integral Domain ◽  
White Paper ◽  
Zero Divisors ◽  
Domain Integral ◽  
Zero Integral

RING, commutative ring, almost a ring, semiring, zero ring, zero property, zero divisors, domain, integral domain, and their underlying definitions are presented in this white paper (knowledge base).

scholarly journals Banach Algebras Which are a Direct Sum of Division Algebras

1988 ◽  
Vol 44 (2) ◽  
pp. 143-145
Author(s):  
Antonio Fernandez Lopez ◽  
Eulalia Garcia Rus
Keyword(s):  
Finite Number ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Division Algebras ◽  
Direct Sum

AbstractIn this note it is proved that a (real or complex) semiprime Banach algebra A satisfying xAx = x2Ax2 for every x ∈ A is a direct sum of a finite number of division Banach algebras.

Some properties of a family of quotients of the Rees algebra

2019 ◽  
Vol 18 (06) ◽  
pp. 1950113 ◽  
Author(s):  
Elham Tavasoli
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Proper Ideal ◽  
Rees Algebra ◽  
Finitely Generated ◽  
Flat Ideal

Let [Formula: see text] be a commutative ring and let [Formula: see text] be a nonzero proper ideal of [Formula: see text]. In this paper, we study the properties of a family of rings [Formula: see text], with [Formula: see text], as quotients of the Rees algebra [Formula: see text], when [Formula: see text] is a semidualizing ideal of Noetherian ring [Formula: see text], and in the case that [Formula: see text] is a flat ideal of [Formula: see text]. In particular, for a Noetherian ring [Formula: see text], it is shown that if [Formula: see text] is a finitely generated [Formula: see text]-module, then [Formula: see text] is totally [Formula: see text]-reflexive as an [Formula: see text]-module if and only if [Formula: see text] is totally reflexive as an [Formula: see text]-module, provided that [Formula: see text] is a semidualizing ideal and [Formula: see text] is reducible in [Formula: see text]. In addition, it is proved that if [Formula: see text] is a nonzero flat ideal of [Formula: see text] and [Formula: see text] is reducible in [Formula: see text], then [Formula: see text], for any [Formula: see text]-module [Formula: see text].

scholarly journals The Centers of Semi-Simple Algebras over a Commutative Ring

1967 ◽  
Vol 30 ◽  
pp. 285-293 ◽  
Author(s):  
Shizuo Endo ◽  
Yutaka Watanabe
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Fundamental Problem ◽  
Simple Algebra ◽  
Complete Answer ◽  
Simple Algebras

Recently A. Hattori introduced in [7], [8], [9] the notion of simple algebras over a commutative ring. Especially, in [9], he examined, as a fundamental problem on simple algebras, whether a directly indecomposable semi-simple algebra is simple or not and gave affirmative answers to this in some particular cases. In this note we shall first prove, as a complete answer to this, that any directly indecomposable semi-simple algebra over a Noetherian ring is simple.

scholarly journals Posets and differential graded algebras

1998 ◽  
Vol 64 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Jacqui Ramagge ◽  
Wayne W. Wheeler
Keyword(s):  
Commutative Ring ◽  
Partially Ordered Set ◽  
Integral Domain ◽  
Ordered Set ◽  
Order Relation ◽  
Graded Algebras ◽  
Partially Ordered ◽  
Differential Graded Algebras ◽  
Finite Posets

AbstractIf P is a partially ordered set and R is a commutative ring, then a certain differential graded R-algebra A•(P) is defined from the order relation on P. The algebra A•() corresponding to the empty poset is always contained in A•(P) so that A•(P) can be regarded as an A•()-algebra. The main result of this paper shows that if R is an integral domain and P and P′ are finite posets such that A•(P)≅A•(P′) as differential graded A•()-algebras, then P and P′ are isomorphic.

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