In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying … See more

Commutative rings
• The prototypical example is the ring of integers with the two operations of addition and multiplication.
• The rational, real and complex numbers are commutative rings of a type called fields 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

Direct product
Let R and S be rings. Then the product R × S can be equipped with the following natural ring structure:
for all r1, r2 in R and … See more

The most familiar example of a ring is the set of all integers ⁠ $${\displaystyle \mathbb {Z} ,}$$ ⁠ consisting of the numbers
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Dedekind
The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. In 1871, Richard Dedekind defined … 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 $${\displaystyle P_{n}=\prod _{i=1}^{n}a_{i}}$$ recursively: let P0 = 1 and let … See more