Difference between revisions of "Ringoid"
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| − | + | A generalization of the notion of an associative ring (cf. [[Associative rings and algebras|Associative rings and algebras]]). Let $ ( \Omega , \Lambda ) $ | |
| + | be the [[Variety of universal algebras|variety of universal algebras]] (cf. also [[Universal algebra|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: | ||
| − | + | $$ | |
| + | ( x _ {1} \dots x _ {n} \omega ) \cdot y = ( x _ {1} y ) \dots | ||
| + | ( x _ {n} y ) \omega ,\ \ | ||
| + | \forall \omega \in \Omega ,\ x _ {i} \in G . | ||
| + | $$ | ||
| − | An ordinary associative ring | + | 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==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)</TD></TR></table> | ||
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====Comments==== | ====Comments==== | ||
Latest revision as of 08:11, 6 June 2020
A generalization of the notion of an associative ring (cf. Associative rings and algebras). Let $ ( \Omega , \Lambda ) $
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 |
How to Cite This Entry:
Ringoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ringoid&oldid=48575
Ringoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ringoid&oldid=48575
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article