“School of Mathematics”
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| Paper IPM / M / 16015 |
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| Abstract: | |
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We generalize the notion of the firm commutative rings to weakly firm ones. A ring is said
to be (weakly) firm if it contains a (weakly) essential prime ideal and the zero-component
of each (weakly) essential prime ideal is (weakly) essential. An (weakly) essential ideal is
one with nonzero intersection with every nonzero (prime) ideal. For a prime ideal P of a
commutative ring A with identity, we denote (as usual) by O_P its zero-component; that
is, the set of members of P that are annihilated by non-members of P. We study rings
in which O_P is a weakly essential ideal whenever P is a weakly essential prime ideal. We
prove that the classical ring of quotients of any ring of this kind is itself of this kind. We
show that direct products of rings of this kind are themselves of this kind. We show that
a ring is not weakly firm (consequently, not firm) if its zero divisor set is an ideal and by
an example show that the class of firm rings is properly contained in the class of weakly
firm rings. We also observe some connections between these type of rings and their total
and zero-divisor graphs via the set of their zero divisors.
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