Proving that factor ring R/I is local if R is local
129025
Quotient ring local if Ring is local
17304
Tensor product of two finitely-generated modules over a local ring is zero
117005
Rules in the ring with coefficients in finite field $\mathbb{F}_p$
1026
Ring homomorphism with $\phi(1_R) \neq1_S$
104233
Is the ring of the polynomials on a finite algebraic set always a product of local rings?
17247
Localisation and prime ideals
100340
Invariant dimension property on a ring without zero divisors
150911
Prove that the ring is Artinian and find a composition series of ideals
74558
Localization of ideals at all primes
68786
"This property is local on" : properties of morphisms of $S$-schemes
36987
A criterion of flat modules
111129
Show that $[K:F]_s = [K:L]_s [L:F]_s$ and $[K:F]_i = [K:L]_i [L:F]_i$.
29359
Localisation isomorphic to a quotient of polynomial ring
121465
$R[x]$ has a subring isomorphic to $R$.
9846
Prove $n-m \mid n^r - m^r\,$ [Factor Theorem, monomial case]
85987
Prove that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw) \subset \mathbb{A}^4$, is not a unique factorization domain
48818
Homomorphism of local rings