$\alpha-$prime ideals

Abstract

Let $R$ be a commutative ring with identity. We give a new generalization to prime ideals called $\alpha-$prime ideal. A proper ideal $P$ of $R$ is called an $\alpha-$prime ideal if for all $a,b$ in $R$ with $ab\in P$, then either $a\in P$ or $\alpha(b)\in P$ where $\alpha \in End(R)$. We study some properties of $\alpha-$prime ideals analogous to prime ideals. We give some characterizations for such generalization and we prove that the intersection of all $\alpha-$primes in a ring $R$ is the set of all $\alpha-$nilpotent elements in $R$. Finally, we give new versions of some famous theorems about prime ideals including $\alpha-$integral domains and $\alpha-$fields.