Math 2.0

definition

Ring

Definition

A ring is a set RR with two binary operations and two elements 0,1∈R0, 1 \in R that satisfy the axioms below. The operations are the addition ++ and the multiplication ⋅\cdot.

  1. (R,+)(R, +) is an abelian group with identity element 00.
  2. The multiplication is associative: (a⋅b)⋅c=a⋅(b⋅c)(a \cdot b) \cdot c = a \cdot (b \cdot c) for all a,b,c∈Ra, b, c \in R.
  3. The element 11 is a multiplicative identity: 1⋅a=a⋅1=a1 \cdot a = a \cdot 1 = a for all a∈Ra \in R.
  4. The multiplication is distributive over the addition: a⋅(b+c)=a⋅b+a⋅ca \cdot (b + c) = a \cdot b + a \cdot c and (b+c)⋅a=b⋅a+c⋅a(b + c) \cdot a = b \cdot a + c \cdot a for all a,b,c∈Ra, b, c \in R.

Remarks

Some authors do not require the element 11. They call the structure of this definition a ring with identity, or a unital ring. The axioms give 0⋅a=a⋅0=00 \cdot a = a \cdot 0 = 0 and (−a)⋅b=−(a⋅b)(-a) \cdot b = -(a \cdot b) for all a,b∈Ra, b \in R. The integers Z\mathbb{Z}, the integers modulo nn and the n×nn \times n real matrices are rings. A ring in which a⋅b=b⋅aa \cdot b = b \cdot a for all a,ba, b is a commutative ring.

Explanations

L1 human

A ring is a place where you can add, subtract and multiply, as with the integers. You cannot always divide. The order of a product can matter: for two matrices AA and BB, ABAB and BABA are often different.

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Sources

  1. Web https://en.wikipedia.org/wiki/Ring_(mathematics), Definition

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