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).

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