Let R be a unital ring with involution. The pseudo core inverses of differences and products two projections are investigated in R. Some equivalent conditions obtained. As applications, the invertibility commutator pq - qp anti-commutator + characterized, where p; q

Let F2 be the free group generated by x and y . In this article, we prove that the commutator of xm and yn is a product of two squares if and only if mn is even. We also show using topological methods that there are infinitely many obstructions for an element in F2 to be a product of two squares. AMS Classification 20F12; 57M07

Let T be a bounded linear operator on X = (Σ `q)p with 1 ≤ q < ∞ and 1 < p <∞. Then T is a commutator if and only if for all non zero λ ∈ C, the operator T − λI is not X-strictly singular.

out, R will represent an associative ring with center Z(R). As usual the commutator xy−yx will be denoted by [x,y]. We will use basic commutator identities [xy,z] = [x,z]y+x[y,z] and [x,yz]= [x,y]z+y[x,z]. Recall that a ring R is prime if aRb = (0) implies that a = 0 or b = 0, and is semiprime if aRa = (0) implies a = 0. An additive mapping x x∗ on a ring R is called involution in case (xy)∗ = ...