Abian's poset and the ordered monoid of annihilator classes in a reduced commutative ring
2014 ◽
Vol 13
(08)
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pp. 1450070
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Let R be a reduced commutative ring with 1 ≠ 0. Then R is a partially ordered set under the Abian order defined by x ≤ y if and only if xy = x2. Let RE be the set of equivalence classes for the equivalence relation on R given by x ~ y if and only if ann R(x) = ann R(y). Then RE is a commutative Boolean monoid with multiplication [x][y] = [xy] and is thus partially ordered by [x] ≤ [y] if and only if [xy] = [x]. In this paper, we study R and RE as both monoids and partially ordered sets. We are particularly interested in when RE can be embedded in R as either a monoid or a partially ordered set.
1976 ◽
Vol 28
(4)
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pp. 820-835
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1979 ◽
Vol 27
(4)
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pp. 495-506
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Keyword(s):
2013 ◽
Vol 12
(04)
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pp. 1250184
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Keyword(s):
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2004 ◽
Vol 2004
(40)
◽
pp. 2145-2147
1964 ◽
Vol 16
◽
pp. 136-148
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