Today we continue with Mathematics, and more specifically the branch of "Discrete Mathematics", in order to get into Rings.
So, without further ado, let's get straight into it!
Rings
Rings are yet another algebraic structure, which is defined as a set together with two binary operations on that set. The binary operations have to satisfy properties similar to those of addition and multiplication of integers. It's thus common to note these binary operations as "+" and "." respectively. Rings are thus denoted as (R, +, .), where R is the set and "+", "." the binary operations on that set.
Ring Axioms
Rings satisfy the following three axioms:
(R, +) is an abelian group, which means that it satisfies:
associativity
commutativity
(additive) identity
(additive) invertibility
(R, .) is an monoid, which means that it satisfies:
associativity
(multiplicative) identity
Multiplication (.) is distributive over addition (+)
Distributivity is the following property:
Commutative Rings
Commutative rings are rings, which additionally satisfy commutativity under the operation of multiplication, basically:
Non-commutative Rings
Rings which don't satisfy commutativity under either of the two binary operations are known as non-commutative rings.
For example, in the case of any ring R and natural number n, the set of all square n x n matrices forms a ring under matrix addition and multiplication, which is non-commutative.
Semirings
Rings which don't satisfy additive invertibility are known as Semirings.
For example, the set of natural numbers together with addition and multiplication is a semiring, because not all elements are invertible under addition.
Subrings
A subset A of a ring (R, +, .) is a subring of R if (A, +) is a subgroup of (R, +) and A is closed under multiplication.
In that context, a set consisting only of 0 or {0} and of course R itself are directly also subrings of R.
Rng
In some definitions rings are defined without the requirement of multiplicative identity. Such structures are more commonly known as rng's.
Partial Order Relations and Sets → Partial Order Relations, POSET (Elements, Max-Min, Upper-Lower Bounds), Hasse Diagrams, Total Order Relations, Lattices
