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.

A doodle is a collection of immersed circles without triple intersections in the 2-sphere. It was shown by second author and Tayler that doodles induce commutator identities (identities amongst commutators) free group. In this paper, we observe idea more closely concentrating on with proper noose systems elementary identities. particular, show there bijection between cobordism classes colored w...

Let R be a prime ring of char R = 2, d a nonzero derivation of R, ρ a nonzero right ideal of R and 0 = b ∈ R such that b[[d2[x, y], [x, y]]n = 0 for all x, y ∈ ρ, n ≥ 1 fixed integer. If [ρ, ρ]ρ = 0 then either bρ = 0 or d(ρ)ρ = 0. Mathematics Subject Classification: 16W25, 16R50, 16N60

Let Mb denote the operation of multiplication by b. Let Hj denote the Hilbert transform performed in the jth coordinate, for j = 1, 2, . . .. Consider the jth order iterated commutator Cb,j = [H1, · · · , [Hj , Mb] · · · ], j = 1, 2, . . . We show that the operator norm of Cb,3 on L (R) is comparable to the norm of b in Chang–Fefferman product BMO(⊗31R 2 +). This fact has some well known equiva...

We prove that the stable commutator lengths of Dehn twists are positive, implying that the natural map from the second bounded cohomology to the ordinary cohomology is not injective, that mapping class groups are not uniformly perfect and that the growth rate of Dehn twists are linear. We then prove that the dimensions of the second bounded cohomology of hyperelliptic mapping class groups and o...