## On Á-Rings: A Generalization Of Integral Domains

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##### Date

2010-08-02

##### Authors

Ayman Badawi

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##### Abstract

<p>Let R be a commutative ring with 1 6= 0 and Nil(R) be its set of nilpotent elements. Recall that a prime ideal of R is called a divided prime if P ½ (x) for every x 2 RnP. The class of rings: H = fR j R is a commutative ring and Nil(R) is a divided prime ideal of Rg has been studied extensively by the speaker(i.e. Badawi). Observe that if R is an integral domain, then R 2 H. Hence H is a much larger class than the class of integral domains. If R 2 H, then R is called a Á-ring.<br />
I wrote the ¯rst paper on Á-rings in 1999 :"Á-pseudo-valuation rings," appeared in Advances in Commutative Ring Theory, 101-110, Lecture Notes Pure Appl. Math. 205, Marcel Dekker, New York/Basel, 1999.</p>

<p>Let R be a commutative ring with 1 6= 0 and Nil(R) be its set of nilpotent elements. Recall that a prime ideal of R is called a divided prime if P ½ (x) for every x 2 RnP. The class of rings: H = fR j R is a commutative ring and Nil(R) is a divided prime ideal of Rg has been studied extensively by the speaker(i.e. Badawi). Observe that if R is an integral domain, then R 2 H. Hence H is a much larger class than the class of integral domains. If R 2 H, then R is called a Á-ring.<br /> I wrote the ¯rst paper on Á-rings in 1999 :"Á-pseudo-valuation rings," appeared in Advances in Commutative Ring Theory, 101-110, Lecture Notes Pure Appl. Math. 205, Marcel Dekker, New York/Basel, 1999.</p>

<p>Let R be a commutative ring with 1 6= 0 and Nil(R) be its set of nilpotent elements. Recall that a prime ideal of R is called a divided prime if P ½ (x) for every x 2 RnP. The class of rings: H = fR j R is a commutative ring and Nil(R) is a divided prime ideal of Rg has been studied extensively by the speaker(i.e. Badawi). Observe that if R is an integral domain, then R 2 H. Hence H is a much larger class than the class of integral domains. If R 2 H, then R is called a Á-ring.<br /> I wrote the ¯rst paper on Á-rings in 1999 :"Á-pseudo-valuation rings," appeared in Advances in Commutative Ring Theory, 101-110, Lecture Notes Pure Appl. Math. 205, Marcel Dekker, New York/Basel, 1999.</p>