Ring

Definition g. A ring is a set \(R\) with two binary operations \(\left( a, b \right) \mapsto a + b\) and \(\left( a, b \right) \mapsto a \cdot b\), referred to as addition and multiplication, such that the following axioms hold:

  1. The set \(R\) is an Abelian group under addition with the additive identity \(0\).
  2. The set \(R\) is a monoid under multiplication with the multiplicative identity \(1\).
  3. Distributivity: Equations \(a \cdot \left( b + c \right) = a \cdot b + a \cdot c\) and \(\left( a + b \right) \cdot c = a \cdot c + b \cdot c\) hold for all \(a, b, c \in R\).

A ring is a commutative ring if it is a commutative monoid under multiplication.

Remark e.

  1. The inverse under addition in a ring is the additive inverse, and the inverse under multiplication is the multiplicative inverse.
  2. We usually omit the symbol for multiplication, and instead of \(a \cdot b\), we write \(ab\).
  3. We can think of subtraction in a ring as addition of the additive inverse.
  4. We can’t think of division as multiplication by the multiplicative inverse since not every element has it’s multiplicative inverse.
  5. Axiom 3. describes the left and right distributive laws.

Example j.

  1. The sets \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\), and \(\mathbb{C}\) are rings with their standard addition and multiplication.
  2. The natural numbers do not form a ring since not all elements have their additive inverse in this set.
  3. The set of integers modulo \(n \in \mathbb{Z}\), denoted \(\mathbb{Z}_n\), is a ring.
definitions algebraic-structures math
Aug 3, 2023