Ringoid
A generalization of the notion of an associative ring (cf. Associative rings and algebras). Let
be the variety of universal algebras (cf. also Universal algebra) of signature \Omega .
The algebra \mathbf G = \{ G , \Omega \cup ( \cdot ) \}
is called a ringoid over the algebra \mathbf G ^ {+} = \{ G , \Omega \}
of the variety ( \Omega , \Lambda ) ,
or an ( \Omega , \Lambda ) -
ringoid, if \mathbf G ^ {+}
belongs to ( \Omega , \Lambda ) ,
the algebra \mathbf G
is a subgroup with respect to the multiplication ( \cdot )
and the right distributive law holds with respect to multiplication:
( x _ {1} \dots x _ {n} \omega ) \cdot y = ( x _ {1} y ) \dots ( x _ {n} y ) \omega ,\ \ \forall \omega \in \Omega ,\ x _ {i} \in G .
The operations of \Omega are called the additive operations of the ringoid \mathbf G , and \mathbf G ^ {+} is called the additive algebra of the ringoid. A ringoid is called distributive if the left distributive law holds also, that is, if
y \cdot ( x _ {1} \dots x _ {n} \omega ) = \ ( y x _ {1} ) \dots ( y x _ {n} ) \omega .
An ordinary associative ring \mathbf G is a distributive ringoid over an Abelian group (and \mathbf G ^ {+} is the additive group of \mathbf G ). A ringoid over a group is called a near-ring, a ringoid over a semi-group a semi-ring, a ringoid over a loop a neo-ring. Rings over rings are also considered (under various names, one of which is Menger algebra).
References
[1] | A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) |
Comments
The term "ringoid" , like groupoid, has at least two unrelated meanings, cf. [a1]–[a3].
References
[a1] | P.J. Hilton, W. Ledermann, "Homology and ringoids. I" Proc. Cambridge Phil. Soc. , 54 (1958) pp. 156–167 |
[a2] | P.J. Hilton, W. Ledermann, "Homology and ringoids. II" Proc. Cambridge Phil. Soc. , 55 (1959) pp. 149–164 |
[a3] | P.J. Hilton, W. Ledermann, "Homology and ringoids. III" Proc. Cambridge Phil. Soc. , 56 (1960) pp. 1–12 |
Ringoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ringoid&oldid=48575