Is there a name for a ring with a distinguished absorbing element (for both addition and multiplication)? There's a standard notion of a monoid with zero in algebra, which is a monoid $M$ having a distinguished element $0$ such that $0m=0=m0$ for all $m\in M$.
Is there a common name for the ring analogue of this notion, that of a "ring" $R$ with a distinguished element $\infty$ with
\begin{align*}
a+\infty=\infty+a          &= \infty,\\
a\cdot\infty=\infty\cdot a &=\infty
\end{align*}
for all $a\in R$? Here we have an element $0_+$, the additive unit, and an element $0=\infty$, the absorbing element for both multiplication and addition ($\infty$). Note in particular that $0\neq 0_+$ except for the trivial ring with a distinguished absorbing element!
Also, I've written "ring" in scare quotes because every element of $R$, except for $\infty$, is assumed to have an additive inverse, but $\infty$ can't have one. Regarding this point, we could speak also of "semirings with distinguished absorbing elements". Do these have a name, too?
 A: I doubt there is a name for this and there isn't much reason to have one because they are very nearly equivalent to ordinary rings.  Let us call such objects "rings with infinity".  Given an ordinary ring $R$, you can form a ring with infinty $R\cup\{\infty\}$ by just adding a new element to be $\infty$.  I claim that in fact every ring with infinity with more than one element is of this form.
So, suppose $R$ is a ring with infinity with more than one element.  This implies $\infty\neq 0$, since if $\infty=0$ then $x=x+0=x+\infty=\infty$ for all $x$.  Similarly, this implies $\infty\neq 1$.  Now suppose $a,b\in R\setminus\{\infty\}$.  Then $a$ and $b$ both have additive inverses, and thus so does $a+b$, so $a+b$ cannot be $\infty$.  Also, note that $ab+(-a)b=0b=0$ so $ab$ cannot be $\infty$ either.  Thus we have shown that $R\setminus\{\infty\}$ contains $0$ and $1$ and is closed under addition and multiplication, from which it easily follows that it is a ring and $R$ is obtained from it by adjoining $\infty$.
So, except for the one-element ring with infinity, rings with infinity are just rings with an extra element added.  Homomorphisms of rings with infinity also have to come from ring homomorphisms, since they preserve elements that have additive inverses and so cannot map non-$\infty$ elements to $\infty$ (unless the codomain has only one element).  So the category of rings with infinity with more than one element is equivalent to the category of rings.  Adding the one-element ring with infinity corresponds to formally adjoining a new terminal object to the category which has no morphisms out of it besides the identity morphism to itself.
(If you consider semirings instead of rings then "semirings with infinity" are much more interesting.  For instance, nontrivial examples include $\{0,1,\dots,n,\infty\}$ for any $n\in\mathbb{N}$ where you declare any sum or product that is greater than $n$ to be $\infty$.  I don't know of any standard name for such things, though.
Also, it seems to me that it would be more natural to require that $0\cdot\infty=0$ rather than $0\cdot\infty=\infty$, for instance in the context of the example $\{0,1,\dots,n,\infty\}$ above.  Note that this then implies that no nonzero element can have an additive inverse by considering $a\cdot\infty+(-a)\cdot\infty$.)
