On Proving the Absence of Zero-Divisors for Semi-Group Rings
Canadian mathematical bulletin, Tome 4 (1961) no. 3, pp. 225-231
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For any semi-group S and any ring Λ with unit 1 (always taken to be distinct from 0, the neutral element of Λ under addition) there is known to exist a ring Λ[S] ⊇S which is a A-bimodule such that (i) S is a sub semi-group of the multiplicative semi-group of Λ[S], (ii) λs = sλ, (iii) λ(st) = (λs)t = s(λt) ( s, t ∊ S and λ∊Λ) and (iv) Sis a Λ -basis of Λ[S]. This ring is uniquely determined by these conditions and is usually called the semi-group ring of S over Λ. It may be described explicitly as consisting of the functions f: S → Λ which vanish at all but finitely many places, with functional addition (f+g) (s) = f(s) + g(s) and convolution (fg) (s) = Σf(u) g(v) (uv = s) as the ring operations, the functional A-bimodule operations (λf) (s) = λf(s) and (fλ) (s) - f(s)λ, and each s ∊ S identified with the characteristic function of { s} with values in Λ.
Banaschewski, Bernhard. On Proving the Absence of Zero-Divisors for Semi-Group Rings. Canadian mathematical bulletin, Tome 4 (1961) no. 3, pp. 225-231. doi: 10.4153/CMB-1961-024-4
@article{10_4153_CMB_1961_024_4,
author = {Banaschewski, Bernhard},
title = {On {Proving} the {Absence} of {Zero-Divisors} for {Semi-Group} {Rings}},
journal = {Canadian mathematical bulletin},
pages = {225--231},
year = {1961},
volume = {4},
number = {3},
doi = {10.4153/CMB-1961-024-4},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1961-024-4/}
}
TY - JOUR AU - Banaschewski, Bernhard TI - On Proving the Absence of Zero-Divisors for Semi-Group Rings JO - Canadian mathematical bulletin PY - 1961 SP - 225 EP - 231 VL - 4 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1961-024-4/ DO - 10.4153/CMB-1961-024-4 ID - 10_4153_CMB_1961_024_4 ER -
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