I'm interested whether there are (i'm sure there are!) some facts about how many zero-divisors might be in rings with certain properties.

For example is there any connection between the cardinality of the ring and the cardinality of the set of its zero divisors? What if it's unitary ring? if it's commutative? if it's boolean?

Especially i'd like to answer the following questions and will be grateful for any help:

1) May the infinite ring have only finite number of zero divisors? If yes, than which additional conditions on ring do we need?

1*) May the infinite non-commutative ring have only finite number of left zero divisors?

2) May there be some left zero divisors and none right zero divisor?

2*) May the ring has finitely many left zero divisors and infinite amount of right zero divisors?

3) Suppose we know that there are $1 \leq n < \infty$ zero divisors. Is there any formula for the number of all finite rings which have exactly $n$ zero divisors? Or at least for the number of non-isomorphic such rings? May be there is some upper bound?

3*) The same question for left (right) zero divisors and the number of finite non-commutative rings

Thanks in advance.

P.S. I use the terminology under which ring may have no unity.

  • $\begingroup$ If there is a left zero divisor in a ring, there is a right zero divisor too, for obvious reasons! $\endgroup$ – Mariano Suárez-Álvarez May 21 '13 at 14:47
  • $\begingroup$ Oh, i'm ashamed :( $\endgroup$ – Igor May 21 '13 at 15:30
  • 2
    $\begingroup$ Here is a link that may be helpful mathoverflow.net/questions/31460/… $\endgroup$ – Kris Williams May 21 '13 at 16:21

Here is one connection: if $R$ is a commutative ring with unity with $k$ zero divisors, $2\leq k < \infty$, then $R$ is a finite ring and $|R|\leq k^2$. This is a result of Ganesan, and further details can be found in his paper and those that cite it:

Ganesan, N. Properties of rings with a finite number of zero divisors. Math. Ann. 157 1964, pp. 215-218.


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