Cotype dimension and cotype chain conditions

2019 ◽  
Vol 18 (01) ◽  
pp. 1950005
Author(s):  
Alejandro Alvarado-García ◽  
Hugo A. Rincón-Mejía ◽  
José Ríos-Montes ◽  
Bertha Tomé-Arreola
Keyword(s):  
Finite Number ◽  
Projective Module ◽  
Chain Conditions ◽  
Direct Sum

Previous results on cotype dimension are generalized from amply supplemented modules to arbitrary ones in order to define the ct-ACC and the ct-DCC conditions. These in turn are used to determine when conat-[Formula: see text] is atomic with a finite number of atoms. Finally, we prove that each noncosingular semiperfect projective module is a direct sum of pairwise coorthogonal [Formula: see text]-atomic modules if and only if so is each noncosingular semiperfect CTS-module with (CT3).

scholarly journals Characterizing rings by a direct decomposition property of their modules

2006 ◽  
Vol 80 (3) ◽  
pp. 359-366 ◽  
Author(s):  
Dinh Van Huynh ◽  
S. Tariq Rizvi
Keyword(s):  
Projective Module ◽  
Direct Decomposition ◽  
Decomposition Property ◽  
Direct Sum ◽  
Continuous Module

AbstractA module M is said to satisfy the condition (℘*) if M is a direct sum of a projective module and a quasi-continuous module. In an earlier paper, we described the structure of rings over which every (countably generated) right module satisfies (℘*), and it was shown that such a ring is right artinian. In this note some additional properties of these rings are obtained. Among other results, we show that a ring over which all right modules satisfy (℘*) is also left artinian, but the property (℘*) is not left-right symmetric.

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.

scholarly journals When every projective module is a direct sum of finitely generated modules

2007 ◽  
Vol 315 (1) ◽  
pp. 454-481 ◽  
Author(s):  
Warren Wm. McGovern ◽  
Gena Puninski ◽  
Philipp Rothmaler
Keyword(s):  
Projective Module ◽  
Finitely Generated ◽  
Direct Sum ◽  
Finitely Generated Modules

scholarly journals A characterization of noetherian rings by cyclic modules

1996 ◽  
Vol 39 (2) ◽  
pp. 253-262 ◽  
Author(s):  
Dinh Van Huynh
Keyword(s):  
Projective Module ◽  
Noetherian Rings ◽  
Noetherian Module ◽  
Direct Sum

It is shown that a ring R is right noetherian if and only if every cyclic right R-module is injective or a direct sum of a projective module and a noetherian module.

scholarly journals RINGS OVER WHICH CYCLICS ARE DIRECT SUMS OF PROJECTIVE AND CS OR NOETHERIAN

2010 ◽  
Vol 52 (A) ◽  
pp. 103-110 ◽  
Author(s):  
C. J. HOLSTON ◽  
S. K. JAIN ◽  
A. LEROY
Keyword(s):  
Projective Module ◽  
Regular Ring ◽  
Artinian Ring ◽  
Von Neumann Regular Ring ◽  
Finitely Generated Module ◽  
Direct Sums ◽  
Von Neumann ◽  
Noetherian Module ◽  
Direct Sum ◽  
Von Neumann Regular

AbstractR is called a right WV-ring if each simple right R-module is injective relative to proper cyclics. If R is a right WV-ring, then R is right uniform or a right V-ring. It is shown that a right WV-ring R is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS (complements are summands, a.k.a. ‘extending modules’) or noetherian module. For a finitely generated module M with projective socle over a V-ring R such that every subfactor of M is a direct sum of a projective module and a CS or noetherian module, we show M = X ⊕ T, where X is semisimple and T is noetherian with zero socle. In the case where M = R, we get R = S ⊕ T, where S is a semisimple artinian ring and T is a direct sum of right noetherian simple rings with zero socle. In addition, if R is a von Neumann regular ring, then it is semisimple artinian.

scholarly journals Rings characterised by semiprimitive modules

1995 ◽  
Vol 52 (1) ◽  
pp. 107-116
Author(s):  
Yasuyuki Hirano ◽  
Dinh Van Huynh ◽  
Jae Keol Park
Keyword(s):  
Direct Summand ◽  
Projective Module ◽  
Injective Module ◽  
Direct Sum

A module M is called a CS-module if every submodule of M is essential in a direct summand of M. It is shown that a ring R is semilocal if and only if every semiprimitive right R-module is CS. Furthermore, it is also shown that the following statements are equivalent for a ring R: (i) R is semiprimary and every right (or left) R-module is injective; (ii) every countably generated semiprimitive right R-module is a direct sum of a projective module and an injective module.

The structure of unique factorization rings

1970 ◽  
Vol 67 (3) ◽  
pp. 535-540 ◽  
Author(s):  
C. R. Fletcher
Keyword(s):  
Finite Number ◽  
Principal Ideal ◽  
Structure Theorem ◽  
Unique Factorization ◽  
Direct Sum ◽  
Unique Factorization Domains

1. Introduction. In (1) we proved that the direct sum of a finite number of unique factorization rings is a unique factorization ring (UFR), and in particular that the direct sum of a finite number of unique factorization domains (UFD's) is a UFR. The converse, however, does not hold i.e. not every UFR can be expressed as a direct sum of UFD's. Here we investigate the structure of UFR's and show that every UFR is a finite direct sum of UFD's and of special UFR's. There is thus a relationship with the structure theorem for principal ideal rings ((2), p. 245).

Nil Subrings of Goldie Rings are Nilpotent

1969 ◽  
Vol 21 ◽  
pp. 904-907 ◽  
Author(s):  
Charles Lanski
Keyword(s):  
Chain Condition ◽  
Chain Conditions ◽  
Direct Sum ◽  
Left Annihilator ◽  
Ascending Chain Condition

Herstein and Small have shown (1) that nil rings which satisfy certain chain conditions are nilpotent. In particular, this is true for nil (left) Goldie rings. The result obtained here is a generalization of their result to the case of any nil subring of a Goldie ring.Definition. Lis a left annihilator in the ring R if there exists a subset S ⊂ R with L = {x∈ R|xS= 0}. In this case we write L= l(S). A right annihilator K = r(S) is defined similarly.Definition. A ring R satisfies the ascending chain condition on left annihilators if any ascending chain of left annihilators terminates at some point. We recall the well-known fact that this condition is inherited by subrings.Definition. R is a Goldie ring if R has no infinite direct sum of left ideals and has the ascending chain condition on left annihilators.

scholarly journals Projective summands in generators

1982 ◽  
Vol 86 ◽  
pp. 203-209 ◽  
Author(s):  
David Eisenbud ◽  
Wolmer Vasconcelos ◽  
Roger Wiegand
Keyword(s):  
Homomorphic Image ◽  
Polynomial Ring ◽  
Dedekind Domain ◽  
Projective Modules ◽  
Finitely Generated ◽  
Positive Direction ◽  
Chain Conditions ◽  
Direct Sum ◽  
Dedekind Domains ◽  
Category Of Modules

An R-module M is a generator (of the category of modules) provided every module is a homomorphic image of a suitable direct sum of copies of M. Equivalently, some M(k) has R as a summand. Except in the last section, all rings are assumed to be commutative, Noetherian domains, and modules are usually finitely generated. In this context generators are exactly those modules that have non-zero free summands locally. Of course, generators can fail to have free summands (e.g., over Dedekind domains), and we ask whether they necessarily have non-zero projective summands. The answer is “yes” for rings of dimension 1, as we point out in § 3, and for the polynomial ring in one variable over a Dedekind domain. In § 1 we show that for 2-dimensional rings the answer is intimately connected with the structure of projective modules. Our main result in the positive direction, Theorem 1.3, grew out of the attempt, in conversations with T. Stafford, to understand the case R = k[x, y]. In § 2 we give examples of rings having generators with no projective summands. The last section contains miscellaneous observations, some of them on rings without chain conditions.

scholarly journals Non-local rings whose ideals are all quasi-injective

1972 ◽  
Vol 6 (1) ◽  
pp. 45-52 ◽  
Author(s):  
G. Ivanov
Keyword(s):  
Finite Number ◽  
Local Rings ◽  
Artinian Rings ◽  
Direct Sum ◽  
Non Local ◽  
Left And Right

A ring is a left Q-ring if all of its left ideals are quasi-injective. For an integer m ≤ 2, a sfield D, and a null D-algebra V whose left and right D-dimensions are both equal to one, let H(m, D, V) be the ring of all m x m matrices whose only non-zero entries are arbitrary elements of D along the diagonal and arbitrary elements of V at the places (2, 1), …, (m, m-l) and (l, m). We show that the only indecomposable non-local left Q-rings are the simple artinian rings and the rings H(m, D, V). An arbitrary left Q-ring is the direct sum of a finite number of indecomposable non-local left Q-rings and a Q-ring whose idempotents are all central.

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