Let $S/R$ be a ring extension and $\beta \in S^*$. Show that for all $\alpha \in R[\beta]\cap R[\beta^{-1}]$ is integral over R.
7903
Prove that $B[x] \cap B[x^{-1}]$ is integral over $B$
24247
Ring Inside an Algebraic Field Extension
98816
How to prove that $\{f+g<a\}=\bigcup_{r\in{\Bbb Q}}(\{f<r\}\cap\{g<a-r\})$?
115920
Showing $R$ is a local ring if and only if all elements of $R$ that are not units form an ideal
118049
Show that $\mathbb{R}^{\mathbb{R}} = U_{e} \oplus U_{o}$
45098
How do we prove the "lying over" property for integral extensions?
7042
Defined integral of $x^{\alpha} (1-x)^{\beta-1}$
141167
What is $\int_{c}^{\infty}\Phi(a+bx) \phi( x ) dx$ for $c\in\mathbb{R}$
41505
Transcendental Extensions. $F(\alpha)$ isomorphic to $F(x)$
35777
$\mathbb{R}[X,Y]/(Y-X^2,Y+X)\cong_\text{Ring} \mathbb{R}\times \mathbb{R}$
28313
Intermediate ring between a field and an algebraic extension.
2575
Showing that $\mathbb{R}$ is connected
112131
Characterizing all ring homomorphisms $C[0,1]\to\mathbb{R}$.
8317
Understanding a proof of "If $R$ is an integral domain, then $R[x]$ is an integral domain as well".
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?
74559
$A$ and $B$ are ideals of a ring ${R}$ such that $A\cap B=\{0\}$. Prove that $st=0$ for every $s\in A, t\in B$.
13983
Show that If $R[X]$ is Euclidean domain then $R$ is a field