Saturday, 26 October 2013

Rings

Here in this post we shall deal with an algebraic structure equipped with two binary  compositions denoted additively and multiplicatively i.e. by +  and   .   and it will be known as Ring .

Definition :-  A non empty set R with two binary compositions to be denoted additively and multiplicatively by symbol  +  and  .  is called a Ring  ( R, + , . )  if it satisfies the following axioms :-

R1 The set R is an abelian group for the additive composition.
R2  Multiplication is binary composition which is associative .
R3  Multiplication is both right  and left distributive with regards to addition .

 So we can understand the Ring as follows

The most familiar example of a ring is the set of all integers, Z, consisting of the numbers
. . . , −4, −3, −2, −1, 0, 1, 2, 3, 4, . . .
There are familiar properties for multiplication and addition of the integers. These properties serve as a model for the axioms for rings. A ring is a set R equipped with two binary operations + and · called addition and multiplication, that map every pair of elements of R to a unique element of R. These operations must satisfy the following properties called ring axioms (the symbol ⋅ is often omitted and multiplication is just denoted by juxtaposition.), which must be true for all a, b, c in R:
  • R is an Abelian group under addition, meaning:
1. (a + b) + c = a + (b + c) (+ is associative)
2. There is an element 0 in R such that 0 + a = a (0 is the zero element)
3. a + b = b + a (+ is commutative)
4. For each a in R there exists −a in R such that a + (−a) = (−a) + a = 0 (−a is the inverse element of a)
  • Multiplication is associative:
5. (ab) ⋅ c = a ⋅ (bc)
  • Multiplication distributes over addition:
6. a ⋅ (b + c) = (ab) + (ac) (left distributivity)
7. (b + c) ⋅ a = (ba) + (ca) (right distributivity)
For many authors, these seven axioms are all that are required in the definition of a ring (such a structure is also called pseudo-ring, or a rng). For others, the following additional axiom is also required:
  • Multiplicative identity
8. There is an element 1 in R such that a ⋅ 1 = 1 ⋅ a = a
Rings which satisfy all eight of the above axioms are sometimes, for emphasis, referred to as unital rings (also called unitary rings, rings with unity, rings with identity or rings with 1). For example, the set of even integers satisfies the first seven axioms, but it does not have a multiplicative identity, and therefore does not satisfy the eighth axiom.
Regarding which convention should be used, Gardner and Wiegandt argue that if one requires all rings to have a unity, then some consequences include the lack of existence of infinite direct sums of rings, and the fact that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable." This article adopts the convention that, unless otherwise stated, a ring is assumed to be unital.
Although ring addition is commutative, so that a + b = b + a, ring multiplication is not required to be commutative; ab need not equal ba. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings.
Some basic properties of a ring follow immediately from the axioms.
  • The additive identity and the additive inverse are unique.
  • The binomial formula holds for any commuting elements (i.e., xy = yx).

Example: Integers modulo 4

Consider the set Z4 consisting of the numbers 0, 1, 2, 3 where addition and multiplication are defined as follows. To avoid possible confusions and to keep the usual notation for the arithmetic operations, we will over-line 0, 1, 2, 3 when considering them in Z4.
  • (Addition) The sum \overline{x} + \overline{y} in Z4 is the remainder of x+y (as an integer) when divided by 4. For example, in Z4 we have \overline{2} + \overline{3} = \overline{1} and \overline{3} + \overline{3} = \overline{2}
  • (Multiplication) The product \overline{x} \cdot \overline{y} in Z4 is the remainder of x\cdot y (as an integer) when divided by 4. For example, in Z4 we have \overline{2} \cdot \overline{3} = \overline{2} and \overline{3} \cdot \overline{3} = \overline{1}.
If x is an integer, the remainder of x when divided by 4 is an element of Z4, and this element is often denoted by "x mod 4", or sometimes \overline{x}, which is coherent with above notation. By checking each axiom, one verifies that Z4 is a ring under these operations. Each axiom follows from the fact that the integers form a ring, and converting the integers to Z4. The additive inverse of any \overline{x} in Z4 is the remainder (-x \mod 4) =\overline{-4}. In other words, we have -\overline{x}=\overline{-x}. For example, in Z4, we have -\overline{3}= \overline{-3} = \overline{1}.
Once one has checked that the ring axioms hold, operations within the ring Z4 become easier to carry out. For example, to compute 3 ⋅ (3 − 1) + 1, one first computes the value within the full set of integers (which is 7), and then converts the result by finding the remainder after dividing by 4, which in this case is 3.

Example: 2-by-2 matrices

Consider the set of 2-by-2 matrices, whose entries are real numbers. This set is written:
\mathcal{M}_2(\mathbb{R}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \bigg|\  a,b,c,d \in \mathbb{R} \right\}
One can check that with the operations of matrix addition and matrix multiplication, this set satisfies the above ring axioms. The element is the multiplicative identity element of the ring. This ring is one of the simplest examples of a non-commutative ring. To see that it is not commutative, consider the following multiplications, which give two matrices A and B such that AB is different from BA:
 \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}  \neq  \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}  = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
One can generalize this construction by replacing the set of real numbers with any ring (not necessarily commutative), and instead of using 2-by-2 matrices, one can use square matrices of any fixed size; see matrix ring.

Rings with extra structure

A ring may be viewed as an abelian group (by using the addition operation), with extra structure. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example:
  • An associative algebra is a ring that is also a vector space over a field K. For instance, the set of n-by-n matrices over the real field R has dimension n2 as a real vector space.
  • A ring R is a topological ring if its set of elements is given a topology which makes the addition map ( + : R\times R \to R\,) and the multiplication map ( \cdot : R\times R \to R\,) to be both continuous as maps between topological spaces (where X × X inherits the product topology or any other product in the category). For example, n-by-n matrices over the real numbers could be given either the Euclidean topology, or the Zariski topology, and in either case one would obtain a topological ring.
It may be noted that for ring there is no necessity for the existence of multiplicative identity and inverse and also multiplicative composition to be commutative . 

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