The separativity problem in terms of varieties and diagonal reduction

We provide two new formulations of the separativity problem. First, it is known that separativity (and strong separativity) in von Neumann regular (and exchange) rings is tightly connected to unit-regularity of certain kinds of elements. By refining this information, we characterize separative regular rings in terms of a special type of inner inverse operation, which is defined via a single identity. This shows that the separative regular rings form a subvariety of the regular rings. The separativity problem reduces to the question of whether every element of the form $(1-aa')bac(1-a'a)$ in a regular ring is unit-regular, where $a'$ is an inner inverse for $a$. Second, it is known that separativity in exchange rings is equivalent to regular matrices over corner rings being reducible to diagonal matrices via elementary row and column operations. We show that for $2\times 2$ invertible matrices, four elementary operations are sufficient, and in general also necessary. Dropping the invertibility hypothesis, but specializing to separative regular rings, we show that three elementary row operations together with three elementary column operations are sufficient, and again in general also necessary. The separativity problem can subsequently be reframed in terms of the explicit number of operations needed to diagonally reduce.