Abstract. Let R be an associative ring with identity 1 and J(R) the Jacobson radical of R. Suppose that m ≥ 1 is a fixed positive integer and R an m-torsion-free ring with 1. In the present paper, it is shown that R is commutative if R satisfies both the conditions (i) [xm, ym] = 0 for all x, y ∈ R\J(R) and (ii) [x, [x, ym]] = 0, for all x, y ∈ R\J(R). This result is also valid if (ii) is repla...

We establish commutativity theorems for certain classes of rings in which every invertible element is central, or, more generally, in which all invertible elements commute with one another. We prove that if R is a semiexchange ring (i.e. its factor ring modulo its Jacobson radical is an exchange ring) with all invertible elements central, then R is commutative. We also prove that if R is a semi...

Introduction ............................................................................................. 1716 I. Background ................................................................................... 1718 A. Protection of Public Health in Jacobson v. Massachusetts ... 1718 B. Strict Scrutiny and the Fundamental Right ............................ 1721 C. Individual Liberty and the Righ...

Commutative Hilbertian Frobenius algebras are those commutative semigroup objects in the monoidal category of Hilbert spaces, for which adjoint multiplication satisfies compatibility relation, that is, this “comultiplication” is a bimodule map. In note we show relation forces operators to be normal. We then prove these have strong Wedderburn decomposition where (ortho)complement Jacobson radica...

Jacobson, M.Z., M.A. Delucchi, M.A. Cameron, and B.V, Mathiesen, Matching demand with supply at low cost among 139 countries within 20 world regions with 100% intermittent wind, water, and sunlight (WWS) for all purposes, Renewable Energy, 123, 236-248, 2018, https://doi.org/10.1016/j.renene.2018.02.009 , https://web.stanford.edu/group/efmh/jacobson/Articles/I/CombiningRenew/WorldGridIntegratio...