# Algebraic definition of ringoid

Category theory has the concept of a groupoid and this is a different concept from the use of the word groupoid to refer to a magma.

Wikipedia gives an algebraic definition of this concept of groupoid as a set with a partial binary function.

A ringoid is described as a “ring with many objects”.

What is an algebraic definition of a ringoid along the same lines as the algebraic definition of a groupoid given by Wikipedia? Is it just a ring in which the addition is only partially defined, or is the multiplication also only partially defined?

• What is any definition of a ringoid? – Chris Eagle Mar 7 '13 at 12:14
• According to Wolfram, it's this: mathworld.wolfram.com/Ringoid.html – mdp Mar 7 '13 at 12:17
• The Wolfram definition is, I think, not referring to the category notion of ringoid but is comparable to the use of the term groupoid to refer to a magma. – Snor Mar 7 '13 at 12:20

## 1 Answer

If you think of a ring as a monoid in the category of abelian groups, you might say a ringoid is a category in the category of abelian groups, or in the category of commutative groupoids.

Groupoids are less a product of loosening axioms as to their meaning in categories, such as in homotopy theory. This is why people don't generally study quasigroupoids. EDIT per OP EDIT: So, it's an accident that the definition of a groupoid corresponds to weakening of the closure axiom - the idea is that weakening the group operation to a partial operation, one where not all members can be multiplied, is one way of saying that, while morphisms are all invertible, not every morphism is an isomorphism. It's better to think of a groupoid as a graph where edges only come in reversing pairs.

EDIT: Hence, nLab defines a ringoid as a category enriched over the category of Abelian groups.

• But if you want a category-based generalization of rings with an interesting definition in terms of weakened algebraic axioms, you might look at hyperrings and hypergroups. These don't send elements to elements, but elements or sets to sets. – Loki Clock Mar 7 '13 at 13:07
• What does "while morphisms are all invertible, not every morphism is an isomorphism" mean? What is the difference between being invertible and being an isomorphism!? – user337830 Nov 25 '16 at 10:52
• @user337830 That should say automorphism. – Loki Clock Jun 1 '17 at 13:37