A ring is a set R equipped with two binary operations + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms
1. R … See more

Dedekind
The study of rings originated from the theory of polynomial rings and the theory of See more

Products and powers
For each nonnegative integer n, given a sequence $${\displaystyle (a_{1},\dots ,a_{n})}$$ of n elements of R, one can define the product See more

Direct product
Let R and S be rings. Then the product R × S can be equipped with the following natural ring structure: See more

The most familiar example of a ring is the set of all integers $${\displaystyle \mathbb {Z} ,}$$ consisting of the numbers
$${\displaystyle \dots ,-5,-4,-3,-2,-1,0,1,2,3,4,5,\dots }$$
The axioms of a ring were elaborated as a generalization of … See more

Commutative rings
• The prototypical example is the ring of integers with the two operations of addition and multiplication. See more

The concept of a module over a ring generalizes the concept of a vector space (over a field) by generalizing from multiplication of … See more