# What is commutative?

• (adj): Of a binary operation; independent of order.
Example: "As in e.g. "a x b = b x a"

### Some articles on commutative:

Semiring
... addition and multiplication, such that (R, +) is a commutative monoid with identity element 0 (a + b) + c = a + (b + c) 0 + a = a + 0 = a a + b = b + a (R ... The difference between rings and semirings, then, is that addition yields only a commutative monoid, not necessarily a commutative group ... A commutative semiring is one whose multiplication is commutative ...
Combinatorial Commutative Algebra
... Combinatorial commutative algebra is a relatively new, rapidly developing mathematical discipline ... lies at the intersection of two more established fields, commutative algebra and combinatorics, and frequently uses methods of one to address problems arising in the other ... can be formulated purely in geometric terms, the methods of the proof drew on commutative algebra techniques ...
Associated Prime
... In commutative algebra the usual definition is different, but equivalent if R is commutative, an associated prime of M is a prime ideal which is of the form for ... In a commutative ring R, an associated prime of M is called an isolated prime if it does not properly contain another associated prime of M ... A finitely generated module over a commutative Noetherian ring is coprimary if and only if it has at most one associated prime ...
Category Of Rings - Subcategories - Category of Commutative Rings
... The category of commutative rings, denoted CRing, is the full subcategory of Ring whose objects are all commutative rings ... of the central objects of study in the subject of commutative algebra ... Any ring can be made commutative by taking the quotient by the ideal generated by all elements of the form (xy − yx) ...
Specific Types of Mathematical Diagrams - Commutative Diagram
... In mathematics, and especially in category theory a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows ... Commutative diagrams play the role in category theory that equations play in algebra ...