All Questions

Recall that, if $R$ is a commutative ring, then a suitably finite $R$-module $M$ is projective if and only if the localization $M_\mathfrak{m}$ is a direct sum of finitely many copies of $R_\mathfrak{...

Let $R=\mathbb{Q}[X,Y]$ be the polynomial ring of two commuting variable. Let $S$ be the multiplicative subset of $R$ generated by homogeneous linear polynomials. Let also $\widehat{R}$ be the ring of ...

Let S be a multiplicative set in a ring R. We can see that every finitely generated $(S^{-1})R$-module is a localization of a finitely generated R-module. Then, more generally, is every finitely n-...

Let $A$ be a commutative ring and $S\subset A$ a subset. A localization of $A$ at $S$ is defined as a ring morphsim $A\to A[S^{-1}]$ which is initial with respect to inverting $S$. Similarly, a ...

Let $R$ be a commutative ring with unity. Let us say that an $R$-module $M$ satisfies property $\mathcal P$ if every submodule of $M$ is a homomorphic image of $M$. Can we characterize all ...

Suppose $M, N$ are two $R$-modules (I had in mind the group ring $R=\mathbb{Z}[G]$ for a finite group $G$). By localizing at a prime $p$ I mean $M_{(p)}\cong M\otimes_R R_{(p)}$. If $M$ and $N$ are ...