Ring

algebra

Definition

Ring

A ring $(R,+,\cdot )$ is an algebraic structure that fulfils the following axioms:

1. $(R,+)$ is an Abelian group
2. $(R,\cdot )$ is a semigroup
3. Left Distributivity: $\forall x,y,z\in R:x\cdot (y+z)=x\cdot y+x\cdot z$
4. Right Distributivity: $\forall x,y,z\in R:(x+y)\cdot z=x\cdot z+y\cdot z$

Depending on the context, this structure can be called “Rng” instead of “Ring” because Rngs don’t postulate a multiplicative identity ($1$, neutral element).

Variants

There are several extensions of the definition of a ring, such as the following:

Commutative Ring

Definition

Commutative Ring

A commutative ring is a ring with $(R,\cdot )$ being commutative.

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Ring with 1

Definition

Ring with 1

A Ring with 1 is a ring with a multiplicative identity ($1$, neutral element).

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