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The answer to this quite beautiful question is that there does exist a commutative ring $R$ with $R\cong R[X,Y]$ but $R\not\cong R[X]$. Let $F$ be a field, and take $$ R=F[x_i,y_i,r_i\ (i\geq 0)] $$ …

Is there an abelian group $A$ with $A\not\cong A\oplus A\cong A\oplus A\oplus A\oplus\cdots$ (a direct sum of countably many copies of $A$)? Edited to add: As no answers are forthcoming, does anyone …

There are three answers. Throughout let $qcl(F)$ be the quadratic closure of a field $F$ inside $\mathbb{C}$. Part 1: Yes there is a quadratically closed field strictly between $qcl(\mathbb{Q})$ and …

My general rule of thumb is to ask myself the following questions: (1) If I pretend that my paper was written by someone else, and I hadn't seen it before, and I want to verify its accuracy, would …

In rings: Let $R$ be a ring where idempotents lift modulo the Jacobson radical $J(R)$. Any countable set of orthogonal idempotents in $R/J(R)$ lifts to an orthogonal set of idempotents; but this fai …

Let $R$ be the subring of $\prod_{i=1}^{\infty}\mathbb{Q}$ of sequences which are eventually constant. This ring has the "obvious" maximal ideals $M_i$ of sequences which are zero in the $i$th coordi …

Let $\pi:R\to R/I$ be the natural projection map. This is an $R$-module homomorphism (as well as a ring homomorphism). If $R/I$ is projective, then this map splits. Call such a splitting $\varphi:R …

Factorization in the ring $\mathbb{Z}[x]/(x^2+1)\mathbb{Z}[x]\cong \mathbb{Z}[i]$ is well known. For instance, $5$ and $13$ (and any prime $\equiv 1\pmod{4}$) are no longer prime. The factorization o …

On question 1: Let $A=\mathbb{F}_2[a,b,d\ :\ a^2=b^2,ad=a,bd=b]$. The element $a$ is not prime since $A/(a)$ has a nonzero nilpotent element $\overline{b}$. The element $a$ also is not irreducible s …

(This is only a comment.) If you instead look at the family $\{x^d-x+n\}$, then it is known that there are infinitely many irreducible elements (for each positive $n$). It suffices to show that $f(x …

Suppose $F$ is a field, and $F_1, F_2$ are two extension fields of $F$. Is it always the case that there is a field $L$, containing three subfields $F, K_1, K_2$ and two ring isomorphisms $\varphi_{i …

I've figured out how to do the finite graph case, in the negative. The ordering of quadratically closed fields is not dense in that case either. Throughout let $qcl(F)$ denote the quadratic closure …

Pete, here is another nice example (shown to me by Bergman many years ago) answering question 1 in another way. Let $F$ be a field, let $R=F[x]$, and let $M=F[[x]]$ (which is an $R$-module in the obv …

I believe that the answer is yes. Let $A$ be a commutative ring with $1$, such that any non-empty set of ideals has a maximal element. For each ideal $I\leq A[[x]]$, and each $n\in \mathbb{N}$, defi …

I believe that the answer is yes. Put $c:=g(x_1,\ldots, x_n)$, which is irreducible in the UFD $R:=k[x_1,\ldots, x_n]$. Assume $f$ is irreducible, but also assume by way of contradiction that $f(c)= …