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Ring

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A ring is an algebraic structure comprised of a set paired with two operations on the set, which are designated as addition (+) and multiplication (\cdot). As a group can be conceptualized as an ordered pair of a set and an operation, \left(G,\cdot\right), a ring can be conceptualized as an ordered triple \left(R,+,\cdot\right).

A set with addition and multiplication, \left(R,+,\cdot\right), is a field if and only if it satisfies the following properties:

  1. Commutativity of addition — For all a, b \in R, a + b = b + a;
  2. Associativity of both addition and multiplication — For all a, b, c \in R, \left(a + b\right) + c = a + \left(b + c\right) and \left(a \cdot b\right) \cdot c = a \cdot \left(b \cdot c\right);
  3. Additive IdentityThere exists a "zero" element, 0 \in R, called an additive identity, such that a + 0 = a = 0 + a, for all a \in R;
  4. Additive Inverses — For each a \in R, there exists a b \in R, called an additive inverse of a, such that a + b = 0;
  5. Distributive property — For all a, b, c \in R, a \cdot \left(b + c\right) = a \cdot b + a \cdot c and \left(b + c\right) \cdot a = b \cdot a + c \cdot a;
  6. Closure of addition and multiplication — For all a, b \in R, a + b \in R and a \cdot b \in R.

A prototype of these kind of structures in the ring of the integers, (\mathbb{Z},+,\cdot).

We will often abbreviate the multiplication of two elements, a \cdot b, by juxtaposition of the elements, ab. Also, when combining multiplication and addition in an expression, multiplication takes precedence over addition unless the addition is enclosed in parenthesis. That is, a + bc + d = a + \left(bc\right) + d.

We can also denote -a as the additive inverse of any a \in R. Furthermore, we can define an operation, called subtraction by a - b = a + \left(-b\right).

Optional Properties

Optionally, a ring R may have additional properties:

  • We define R to be a commutative ring if the multiplication is commutative: a \cdot b = b \cdot a for all a,b\in R;
  • We define R to be a ring with unity if there exists a multiplicative identity 1\in R: 1\cdot a = a = a \cdot 1 for all a \in R;
    • Furthermore, a commutative ring with unity R is a field if every element except 0 has a multiplicative inverse: For each non-zero a \in R, there exists a b\in R such that a\cdot b = b \cdot a = 1;
  • We define R to be an integral domain if it is commutative, has unity, and the zero product rule holds: a \cdot b = 0 implies either a  = 0 or b = 0 for all a,b \in R.

Important Results

From the given axioms for a ring R, it can be deduced that:

  • \left(R,+\right) is an abelian group;
  • 0 \cdot a = 0, for all a \in R;
  • a\left(-b\right) = \left(-a\right)b = -\left(ab\right), for all a, b \in R;
  • \left(-a\right)\left(-b\right) = ab, for all a, b \in R;
  • If R has unity 1:
    • \left(-1\right) \cdot a = -a, for all a \in R;
    • \left(-1\right) \cdot \left(-1\right) = 1;
  • Multiplication distributes over subtraction.

Substructures, morphisms and quotients

Examples and results

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