On $\phi$-pseudo-Krull rings
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The purpose of this paper is to introduce a new class of rings that is closely related to the class of pseudo-Krull domains. Let $\mathcal{H} = \{R \,|\, R$ is a commutative ring and $\Nil(R)$ is a divided prime ideal of $R\}$. Let $R\in \mathcal{H}$ be a ring with total quotient ring $T(R)$ and define $\phi : T(R) \longrightarrow R_{\Nil(R)}$ by $\phi(\frac{a}{b}) = \frac{a}{b}$ for any $a \in R$ and any regular element $b$ of $R$. Then $\phi$ is a ring homomorphism from $T(R)$ into $R_{\Nil(R)}$ and $\phi$ restricted to $R$ is also a ring homomorphism from $R$ into $R_{\Nil(R)}$ given by $\phi(x) = \frac{x}{1}$ for every $x \in R$. We say that $R$ is a $\phi$-pseudo-Krull ring if $\phi(R) = \bigcap R_i$, where each $R_i$ is a nonnil-Noetherian $\phi$-pseudo valuation overring of $\phi(R)$ and for every non-nilpotent element $x \in R$, $\phi(x)$ is a unit in all but finitely many $R_i$. We show that the theories of $\phi$-pseudo Krull rings resemble those of pseudo-Krull domains.
Abstract
The purpose of this paper is to introduce a new class of rings that is closely related to the class of pseudo-Krull domains. Let $\mathcal{H} = \{R \,|\, R$ is a commutative ring and $\Nil(R)$ is a divided prime ideal of $R\}$. Let $R\in \mathcal{H}$ be a ring with total quotient ring $T(R)$ and define $\phi : T(R) \longrightarrow R_{\Nil(R)}$ by $\phi(\frac{a}{b}) = \frac{a}{b}$ for any $a \in R$ and any regular element $b$ of $R$. Then $\phi$ is a ring homomorphism from $T(R)$ into $R_{\Nil(R)}$ and $\phi$ restricted to $R$ is also a ring homomorphism from $R$ into $R_{\Nil(R)}$ given by $\phi(x) = \frac{x}{1}$ for every $x \in R$. We say that $R$ is a $\phi$-pseudo-Krull ring if $\phi(R) = \bigcap R_i$, where each $R_i$ is a nonnil-Noetherian $\phi$-pseudo valuation overring of $\phi(R)$ and for every non-nilpotent element $x \in R$, $\phi(x)$ is a unit in all but finitely many $R_i$. We show that the theories of $\phi$-pseudo Krull rings resemble those of pseudo-Krull domains.
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