Example of finite ring which is not a Bézout ring

Question

A left (or right) Bézout ring is a ring in which any two elements generate a principal left (resp. right) ideal. Assume that we have a finite ring R. Does there exist some classification theorem (necessary and sufficient condition for a finite ring to be a Bézout ring other than just formulating the above definition)? Is the ring of matrices $M_{n \times n}(q)$ over the Galois field $GF(q)$ a Bézout ring and how do you prove that. Are there examples of finite rings (commutative or not) which are not Bézout rings? Are there examples of finite local rings (commutative or not) that are not Bézout rings?

Answer 1

Getting this out of unanswered-limbo

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Is the ring of matrices $_{×}()$ over the Galois field $()$ a Bézout ring and how do you prove that.

Yes it is. Every right ideal and every left ideal are principal, because they are all summands and therefore generated by an idempotent.

Are there examples of finite rings (commutative or not) which are not Bézout rings?

Yes, like $F_2[x,y]/(x,y)^2$ for the field of two elements $F_2$, and also the trivial extension of $\mathbb Z/8\mathbb Z$ by itself.

Are there examples of finite local rings (commutative or not) that are not Bézout rings?

Both rings mentioned above are local.