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Register a new userFinitistic Weak Dimension of Commutative Arithmetical Rings
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Authors
Francois Couchot
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It is proven that each commutative arithmetical ring $R$ has a finitistic weak dimension $leq 2$. More precisely, this dimension is 0 if $R$ is locally IF, 1 if $R$ is locally semicoherent and not IF, and 2 in the other cases.
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In this paper, we introduce and study the $S$-weak global dimension $S$-w.gl.dim$(R)$ of a commutative ring $R$ for some multiplicative subset $S$ of $R$. Moreover, commutative rings with $S$-weak global dimension at most $1$ are studied. Finally, we investigated the $S$-weak global dimension of factor rings and polynomial rings.
Let $mathcal{I}(R)$ be the set of all ideals of a ring $R$, $delta$ be an expansion function of $mathcal{I}(R)$. In this paper, the $delta$-$J$-ideal of a commutative ring is defined, that is, if $a, bin R$ and $abin Iin mathcal{I}(R)$, then $ain J(R)$ (the Jacobson radical of $R$) or $bin delta(I)$. Moreover, some properties of $delta$-$J$-ideals are discussed,such as localizations, homomorphic images, idealization and so on.
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