### Rings

Here in this post we shall deal with an algebraic structure equipped with two binary compositions denoted additively and multiplicatively i.e. by

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 .

The most familiar example of a ring is the set of all integers,

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."

Although ring addition is commutative, so that

Some basic properties of a ring follow immediately from the axioms.

Once one has checked that the ring axioms hold, operations within the ring

**+**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, . . .

**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. (
*a*⋅*b*) ⋅*c*=*a*⋅ (*b*⋅*c*)

- Multiplication distributes over addition:

- 6.
*a*⋅ (*b*+*c*) = (*a*⋅*b*) + (*a*⋅*c*) (left distributivity) - 7. (
*b*+*c*) ⋅*a*= (*b*⋅*a*) + (*c*⋅*a*) (right distributivity)

**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*

**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;*a*⋅*b*need not equal*b*⋅*a*. 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., ).

### Example: Integers modulo 4

Consider the set**Z**_{4}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**Z**_{4}.- (
*Addition*) The sum in**Z**_{4}is the remainder of (as an integer) when divided by 4. For example, in**Z**_{4}we have and

- (
*Multiplication*) The product in**Z**_{4}is the remainder of (as an integer) when divided by 4. For example, in**Z**_{4}we have and

*x*is an integer, the remainder of*x*when divided by 4 is an element of**Z**_{4}, and this element is often denoted by "*x*mod 4", or sometimes , which is coherent with above notation. By checking each axiom, one verifies that**Z**_{4}is a ring under these operations. Each axiom follows from the fact that the integers form a ring, and converting the integers to**Z**_{4}. The additive inverse of any in**Z**_{4}is the remainder In other words, we have For example, in**Z**_{4}, we haveOnce one has checked that the ring axioms hold, operations within the ring

**Z**_{4}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:*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*:### 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*n*^{2}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 ( ) and the multiplication map ( ) 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.

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