$R$ semisimple artinian ring $\Rightarrow\varphi(R)$ is such
130814
Quotient of semi-simple rings
91180
Show $R \setminus S$ is a union of prime ideals
83009
Showing that $R(x)$ is a proper subset of $R((x))$ if $R$ is a field
90314
Maximal ideals in $R[x]$
52992
Proving that P/PJ is a projective right module over R/J
143429
Kernel of ring homomorphism $k[X,Y] \rightarrow k[t^2, t^3]$
142699
Is $R/N(R)$ a faithfully flat $R$-module?
40705
If $M$ is an $R$-module and $I\subseteq\mathrm{Ann}(M)$ an ideal, then $M$ has a structure of $R/I$-module
83812
Artinian ring and faithful module of finite length
39576
Quotient ring being isomorphic to the initial ring
30465
Determining which rings have a module which is not semisimple
25764
Example of a ring such that $R^2\simeq R^3$, but $R\not\simeq R^2$ (as $R$-modules)
142863
Show that every ideal of the matrix ring $M_n(R)$ is of the form $M_n(I)$ where $I$ is an ideal of $R$
22604
Can $R \times R$ be isomorphic to $R$ as rings?
121886
Proving image of $f:D^n\rightarrow \mathbb{R}^n \;$ contains $\;D^n$
121465
$R[x]$ has a subring isomorphic to $R$.
60760
Ring homomorphism and characteristic