$\operatorname{Spec}(R)$ not connected iff $R$ is a product of two rings
16255
If $\mathop{\mathrm{Spec}}A$ is not connected then there is a nontrivial idempotent
78654
An example of a norm which can't be generated by an inner product
70576
Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$
106894
If a subring of a ring R has identity, does R also have the identity?
45523
Can the product of two non invertible elements in a ring be invertible?
35379
Spectrum of infinite product of rings
51898
Show that $\operatorname{Hom}_{R} (R,M) \cong M$
139176
The real line is not homeomorphic to any non-trivial product space
147659
Counterexample to "if $f:A\to A$ induces the identity $\hat f:\operatorname{Spec}(A)\to\operatorname{Spec}(A)$, then $f=\operatorname{id}_A$"
117590
Let I and J be ideals of a ring R. Show by example that the set of products {xy | x ∈ I, y ∈ J} need not be an ideal
142699
Is $R/N(R)$ a faithfully flat $R$-module?
73689
A question about the equivalence relation on the localization of a ring.
3296
Proof That $\mathbb{R} \setminus \mathbb{Q}$ Is Not an $F_{\sigma}$ Set
22245
Show the union of two subrings is generally not a subring
141100
Nontrivial subring with unity different from the whole ring?
63338
Noncommutative failure of IBN: $\,R^n\cong R^m \Rightarrow n - m,$ ring $R$ with maximal ideal $I$ such that $R/I$ is not a division ring?
92599
Prove there generally is no isomorphism between $R[x]/(x^2-a)$ and $R^2$