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Register a new userThe $S$-global dimensions of commutative rings
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Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. First, we introduce and study the $S$-projective dimensions and $S$-injective dimensions of $R$-modules, and then explore the $S$-global dimension $S$-gl.dim$(R)$ of a commutative ring $R$ which is defined to be the supremum of $S$-projective dimensions of all $R$-modules. Finally, we investigated the $S$-global dimension of factor rings and polynomial rings.
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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.
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.
We describe new classes of noetherian local rings $R$ whose finitely generated modules $M$ have the property that $Tor_i^R(M,M)=0$ for $igg 0$ implies that $M$ has finite projective dimension, or $Ext^i_R(M,M)=0$ for $igg 0$ implies that $M$ has finite projective dimension or finite injective dimension.
Let $R$ be a commutative ring. We investigate $R$-modules which can be written as emph{finite} sums of {it {second}} $R$-submodules (we call them emph{second representable}). We provide sufficient conditions for an $R$-module $M$ to be have a (minimal) second presentation, in particular within the class of lifting modules. Moreover, we investigate the class of (emph{main}) emph{second attached prime ideals} related to a module with such a presentation.
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