Tagged Questions
3
votes
1answer
100 views
Rings and idempotent semirings
If $\mathbb{R}$ is the real numbers and x#y=max{x,y} then ($\mathbb{R}$,#,+) is a semiring where ($\mathbb{R}$,#) is a semigroup and + distributes over #.
If you have a set R with three distinct ...
6
votes
1answer
164 views
Is the additive semigroup of natural numbers the multiplicative semigroup of a ring?
$\mathbb N$ will denote the set $\{0,1,2,\ldots\}.$ The semigroup $(\mathbb N,+)$ doesn't have a zero element. $\mathbb N^0$ will denote the semigroup $\mathbb N$ with zero adjoined, that is the set ...
6
votes
0answers
114 views
Is there an upper bound to the number of rings that can be obtained from a semigroup with zero by defining an additive operation?
Let $\mathscr S$ be the class of all semigroups with zero. For $(S,\times,0)\in\mathscr S,$ I want to count additive operations $+$ on $S$ such that $(S,+,\times,0)$ is a ring (possibly without ...
5
votes
1answer
101 views
Must a pseudoinverse of a von Neumann regular element be regular?
Background
Let $R$ be a ring or a semigroup. We say that $x\in R$ is a von Neumann regular element of $R$ if there exists $y\in R$ such that
$$xyx=x.$$
Any $y\in R$ satisfying the above equation is ...
5
votes
2answers
169 views
If the product of two idempotents is idempotent, must the two idempotents commute?
It is a basic fact that when two idempotents $e,f$ in a semigroup $S$ commute, then $ef$ is an idempotent. Is the converse true? Is it true for idempotents in rings?
