Ring
Whereas a group is defined by a set and a single operation, a ring is defined by a set and two operations. Given a set R and operations * and +, we say that (R, +, *) is a ring if it satisfies the following properties:
(R, +)is an abelian group- For any x and y ∈ R,
x * y∈ R. - For any x, y, and z ∈ R,
(x * y) * z = x * (y * z). - For any x, y, and z ∈ R,
x * (y + z) = x * y + x * zand(x + y) * z = x * z + y * z
Notes on rings:
- A ring necessarily has an identity under
+(called the additive identity), typically referred to as zero, 0). - A ring does not necessarily have an identity under
*(called the multiplicative identity). - A ring is not necessarily commutitive under
*. - A ring does not necessarily satisfy the inverse property under
*. - There is not necessarily any special property of the additive identity under multiplication.
A few other types of rings:
- We say that
(R, +, *)is a ring with unity if it has a multiplicative identity, referred to as one, or 1. - We say that
(R, +, *)is a commutative ring if it satisfies the commutative property under multiplication, that isx * y = y * x∀ x,y ∈ R.
Examples of rings
(Z, +, *)- The set of square square matrices of a given size are a ring.
AdjoinedUnitRing
Defined in this file. This is for the case where your Ring[T] is a Rng (i.e. there is no unit). see this page.
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