ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

2. Algebra

1991 MS Classification: 16 Artamonova, Irena, Moscow State University, Russia We consider semirings with 1 ≠ 0 Definition. The mapping α : S → R is called an m-isomorphism if α is an isomorphism of the multiplicative semigroups of the semirings S and R, i. e. α : (S, ·) ≅ (R, ·). Definition. A semiring S is called an UA-semiring (a semiring with unique addition) if any misomorphism α : (S, ·) ≅ (R, ·) of the semiring S onto an arbitrary semiring R is an isomorphism of semirings. In the class of boolean semirings we have: Theorem . Any nontrivial boolean ring (which is an UAring) ⟨R, ·, +⟩ is not an UA-semiring. The next theorem has many useful corollaries. Theorem . Let S be a semiring with idempotents e and f such that e^{2} = e, f^{2} = f, ef = fe = 0 and e + f = 1 , with the following conditions: ∀s ∈ S exesf = eyesf ⇒ exe = eye; fxfse = fyfse ⇒ fxf = fyf for x, y ∈ S. Then S is an UA-semiring. Thus such semirings as M_{n}(S) and T_{n}(S) over a semiring S are the UAsemirings. Artamonova I. I., On UA-semirings, Fundamentalnaja i prikladnaja matematika, 1997, N 4, V 3.