Elementary divisor rings and finitely presented modules

Larsen, Max D.; Lewis, W. J.; Shores, Thomas S.

doi:10.1090/s0002-9947-1974-0335499-1

Cited by 72 publications

(18 citation statements)

References 13 publications

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“…Consider the rings in which zero is an avoidable element. By [3], any regular ring is a ring in which zero is an adequate element. By Theorem 1, we have the following result.…”

Section: Definitionmentioning

confidence: 99%

On feebly compact inverse primitive (semi)topological semigroups

Zabavsky¹,

Gatalevych²

2015

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“…Consider the rings in which zero is an avoidable element. By [3], any regular ring is a ring in which zero is an adequate element. By Theorem 1, we have the following result.…”

Section: Definitionmentioning

confidence: 99%

On feebly compact inverse primitive (semi)topological semigroups

Zabavsky¹,

Gatalevych²

2015

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“…It is shown in [7,Theorem 4] that in an adequate domain every non-zero prime ideal is contained in a unique maximal ideal. For the proofs of other assertions in this paragraph, see [2,Theorems 3.18 and 3.19], and [14].…”

Section: Properties Of E(r)mentioning

confidence: 99%

Residue class rings of real-analytic and entire functions

Golasiński

Henriksen

2006

Colloq. Math.

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Abstract. Let A(R) and E(R) denote respectively the ring of analytic and real entire functions in one variable. It is shown that if m is a maximal ideal of A(R), then A(R)/m is isomorphic either to the reals or a real closed field that is an η 1 -set, while if m is a maximal ideal of E(R), then E(R)/m is isomorphic to one of the latter two fields or to the field of complex numbers. Moreover, we study the residue class rings of prime ideals of these rings and their Krull dimensions. Use is made of a classical characterization of algebraically closed fields due to E. Steinitz and techniques described in L. Gillman and M. Jerison's book on rings of continuous functions.

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“…The following wellknown lemma will be useful later: For Λί a finitely generated R-module, we denote by μ(R,M) the minimum number of generators required for Λf over R. matrix (α iy ) in which α« divides a i+UM9 R is an elementary divisor ring. It has recently been shown [7] that a ring is an elementary divisor ring if and only if every finitely presented module is a direct sum of cyclics. We will see later, in (3.4), that the conditions of (1.5) also have a matrix-theoretic formulation.…”

Section: Preliminariesmentioning

confidence: 99%

“…In order to prove that R is an elementary divisor ring it will suffice, by the proof of [7,Theorem 3.8] Then N(P) is finitely presented, and μ(R P ,N(P) P )^n.…”

Section: The Main Theoremmentioning

confidence: 99%