We establish a link between abelian regular subgroup of the affine group, and commutative, associative algebra structures on the underlying vector space that are (Jacobson) radical rings. As an application, we show that if the underlying field has positive characteristic , then an abelian regular subgroup has finite exponent if the vector space is finite-dimensional, while it can be torsion fre...

We characterize the distributive lattices of Jacobson rings and prove that if a semiring is a distributive lattice of Jacobson rings, then, up to isomorphism, it is equal to the subdirect product of a distributive lattice and a Jacobson ring. Also, we give a general method to construct distributive lattices of Jacobson rings.

A right Johns ring is a Noetherian in which every ideal annihilator. It known that RR the Jacobson radical J(R)J(R) of nilpotent and Soc(R)(R) an essential RR. Moreover, Kasch, is, simple RR-module can be embedded For M∈RM∈R-Mod we use concept MM-annihilator define module (resp. quasi-Johns) as MM such submodule MM-annihilator. called quasi-Johns if any set submodules satisfies ascending chain ...

Theorem 1.1. Let A be a semisimple complex Banach algebra with the following properties: (i) for every closed right ideal R in A, there exists a closed left ideal L such that R∩L= {0} and R+ L= A (each a∈ A can be written in the form a= a1 + a2 with a1 ∈ R, a2 ∈ L); (ii) if a,b in A are such that ab = ba= 0, then ‖a+ b‖2 = ‖a‖2 +‖b‖2. Then A is a commutative proper H∗-algebra [1]. It is easy to...

1. Basics 2 1.1. Commutants 2 1.2. Opposite Rings 3 1.3. Units 3 1.4. Ideals 4 1.5. Modules 4 1.6. Division rings 5 1.7. Endomorphism rings 7 1.8. Monoid Rings 8 1.9. Noetherian and Artinian Rings 11 1.10. Simple rings 15 1.11. Prime ideals and prime rings 17 1.12. Notes 18 2. Wedderburn-Artin Theory 18 2.1. Semisimple modules and rings 18 2.2. Wedderburn-Artin I: Semisimplicity of Mn(D) 20 2.3...

My passion for teaching began early. Throughout high school and college, I tutored English to African immigrants. Later, while student teaching remedial and gifted-and-talented mathematics at a New York City public middle school, I realized the diversity of how students learn. Some students thrive off interactive lectures, while others do not fully digest a lecture until working out examples at...