Is there a (finite) ring with exactly 2 minimal left ideal and exactly 3 minimal right ideal? (rings are assumed to have identity)
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Let $M$ be the set of all $3\times 2$ matrices over a field $F_2$ with usual addition. Let $P$ be the $2\times 3$ matrix consisting of $1$'s. Define a multiplication: $A*B= APB$. Then we get a ring (this is so-called Munn algebra) with required properties (a minimal left ideal is generated by a column).
In such a way you can obtain a ring with $m$ minimal left ideals and $n$ minimal right ideals.
Addendum (Thanks to Manny Reyes) $M$ doesn't contain an identity element, but we can join an identity $e$ to $M$, i.e. consider the ring $M_1$ of pairs $(a,me)$ ($a\in M, m\in F_2$) with multiplication $(a,me)*(b,ne)=(a*b+na+mb,mne)$. Then minimal ideals of $M_1$ are the same of $M$.
answered Sep 26, 2013 at 16:37
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$\begingroup$ This is a very clever idea, but this ring doesn't seem to have an identity element. Such an element $X \in M$ would need to be a $3 \times 2$ matrix satisfying $PX = I_2$ and $XP = I_3$. The latter is impossible by rank considerations. Am I missing something? $\endgroup$ Commented Sep 26, 2013 at 19:11
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$\begingroup$ @Manny Reyes: Thank you very much, I forgot about the identity. I will correct the answer. $\endgroup$ Commented Sep 26, 2013 at 19:43
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$\begingroup$ +1 Nice. What can you tell about the form of maximal (left or right) and prime ideals of $M$ (or $M_1$)? $\endgroup$– user37834Commented Sep 26, 2013 at 20:48
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$\begingroup$ @Silvi: There is an analogy with usual matrix rings over a field: if we consider a minimal left ideal as the additive group, it is the direct sum of the additive groups of the field. $\endgroup$ Commented Sep 26, 2013 at 21:05
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$\begingroup$ So what are the maximal left ideals of $M$? $\endgroup$– user37834Commented Sep 28, 2013 at 19:23
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Consider the ring $R = \Bbb{Z}_2<a,b \; | \; a^3 = b^2 = ba = a^2b = 0>$ ($a,b$ are non-commuting indeterminate). Then every element of $R$ is a linear combination of $1,a,b,a^2,ab$. Therefore $|R| = 32$, also $R$ is a local ring and satisfies the asked conditions.