# Tagged Questions

Questions about commutative rings, their ideals, and their modules.

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### direct limit of finitely generated submodule

if $A$ is a module,then the family fin($A$) of all the finitely generated submodules of $A$ is a directed set and direct limit of$M_i$ is isomorphic to$A$. for prove this needed to define to injection ...
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### Is $\operatorname{Hom}_R(R/m,R/(x_1,…,x_d))$ isomorphic to $R/m$?

Let $(R,m)$ be a local ring. Let $x_1,...,x_d$ be a maximal $R$-sequence. Is $\operatorname{Hom}_R(R/m,R/(x_1,...,x_d))$ isomorphic to $R/m$?
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### Question on homogeneous ideal in a graded ring

In an $\Bbb{N}$-graded ring $R=\bigoplus_nR_n$, an element is called homogenous (of degree $n$) if it is contained in $R_n$. An ideal is called homogenous if it is generated by homogenous elements. ...
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### Algebraic Ideal and affine varieties equality

Let $I=\langle x^2-y-4\rangle$ and $G= \langle x^2+y-4\rangle$ be two ideals in the polynomial ring $\mathbb{C}[x,y]$. As obviously $V(I)=V(G)$, what can we conclude for $I$ and $G$? How they are ...
### a f.g., projective, non free $R-$module [duplicate]
I know that if $R$ is a PID ring, then a projective $R-$module is free. Now, i want an example of a f.g., projective, non free $R-$module where $R$ is a non PID ring.
I stumbled across something that I really couldn't really figure out. So suppose you have a morphism of affine algebraic sets: $f: X \rightarrow Y$ and the corresponding coordinate ring morphisms: \$f'...