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Paper   IPM / M / 16015
School of Mathematics
  Title:   On weakly firm commutative rings and their connection to the total and zero-divisor graphs
  Author(s):  Amir Masoud Rahimi (Joint with E. Mehdi-Nezhad)
  Status:   To Appear
  Journal: Palestine J. Math.
  Supported by:  IPM
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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