Prove: If $\mathit R$ is a commutative ring with unity and $\mathit I=(x)\subseteq R[x]$, then $R[x] / (x)\cong R$
55902
Show $R[x]/(x)\cong R$
119067
Basic Ring Theory Question involving the unit element
78626
Let $R$ be a commutative ring. If $R[X]$ is a principal ideal domain, then $R$ is a field.
121750
Let $R$ be a commutative, unital ring and let $I$ be an ideal of $R.$ Prove that if $I$ is maximal then $R/I$ is a field.
85913
R is commutative ring with identity & define $\circ$ on $R$ by for any $a,b \in R$ $a \circ b=a+b-ab$ Prove the following
80961
Prove that a commutative ring $F$ is a field iff $F[x]$ is a Principal Ideal Domain
111493
Division algorithm for polynomials in R[x], where R is a commutative ring with unity.
24259
Quotient ring of $T/I$
114343
If $I$ and $J$ are ideals in a ring $R$ with $1$ such that $I+J = R$, show that $I^m$ and $J^n$ are co-maximal for all $m,n \in \mathbb{N}$
65224
Prove that if $a^2 = a$ for all elements of the ring $R$, then $R$ has characteristic 2. Is the converse of the statement true?
57031
Ring of fractions of $R$ is integral over $R$
6896
If $A/I \cong A/J$ as rings and $I\subseteq J,$ then $I=J.$
73689
A question about the equivalence relation on the localization of a ring.
69649
If an ideal contains the multiplicative identity, then it is the whole ring
36996
If $x^{2}-x\in Z(R)$ for all $x\in R$, then $R$ is commutative.
115920
Showing $R$ is a local ring if and only if all elements of $R$ that are not units form an ideal
131082
Proof that $\mathbb R/[0,1]$ and $\mathbb R$ are homeomorphic