Ring

Definition of a Ring:

  • Set and Operations: A ring (R,+,) consists of a set R equipped with two binary operations, addition + and multiplication , satisfying the following properties:

    1. Additive Group: (R,+) is an abelian group.

      • Closure under addition: For all a,b in R, a+b is in R.
      • Associativity: For all a,b,c in R, (a+b)+c=a+(b+c).
      • Identity element (zero): There exists an element 0 in R such that for any a in R, a+0=0+a=a.
      • Inverse element: For each a in R, there exists an element a in R such that a+(a)=(a)+a=0.
      • Commutativity: For all a,b in R, a+b=b+a.
    2. Multiplication: The multiplication operation is associative.

      • Distributive laws hold: For all a,b,c in R, a(b+c)=ab+ac and (a+b)c=ac+bc.

Key Concepts:

  • Zero and Unity: The additive identity element is denoted as 0, and if a multiplicative identity element exists in the ring, it's called unity, often denoted as 1.

  • Commutative Ring: If the multiplication operation in a ring is commutative (i.e., ab=ba for all a,b in R), the ring is called commutative.

  • Ring with Unity: A ring R is said to have a unity if there exists an element 1 in R such that for any a in R, a1=1a=a.

  • Zero Divisors: Elements a,b in a ring R are zero divisors if a0, b0, and ab=0. Rings without zero divisors are called integral domains.

Examples:

  • Integers: The set of integers Z forms a ring under addition and multiplication. It is both commutative and has unity.

  • Polynomial Rings: The set of polynomials with coefficients from a ring forms another ring under polynomial addition and multiplication.

  • Matrix Rings: The set of n×n matrices with entries from a ring R forms a ring under matrix addition and multiplication.

Ring Properties:

  • Associative Property: Both addition and multiplication in a ring are associative.

  • Distributive Property: Multiplication distributes over addition, i.e., a(b+c)=ab+ac and (a+b)c=ac+bc.

  • Ring Homomorphism: A function between two rings f:RS is a ring homomorphism if it preserves addition and multiplication.

  • Subrings: A subset S of a ring R is a subring if S is itself a ring under the same operations as R.

  • Characteristic of a Ring: The smallest positive integer n such that n1=0 in a ring R is called the characteristic of R. If no such n exists, the characteristic is considered to be zero.