Ringoid

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).

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The term "ringoid" , like groupoid, has at least two unrelated meanings, cf. [a1]–[a3].

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