Rings Field and Vector Space

The concept of a group has its origin in the set of mappings or permutations of a set unto itself.... Click here to read more

Some basic elementary properties of a ring can be illustrated with the help of the following theorem, and these properties... Click here to read more

Example 1: A Gaussian integer is a complex number , where and are integers. Show that the set of Gaussian... Click here to read more

1. Commutative Rings A ring is said to be a commutative if the multiplication composition in is commutative, i.e. 2.... Click here to read more

Cancellation Laws in a Ring We say that cancellation laws hold in a ring if and where are in .... Click here to read more

Let be a ring. A non–empty subset of the set is said to be a subring of if is closed... Click here to read more

Theorem: The intersection of two subrings is a subring. Proof: Let and be two subrings of ring . Since and... Click here to read more

Ideals: Let be any ring and be a subring of , then is said to be a right ideal of... Click here to read more

Integral Domain: A commutative ring with unity is said to be an integral domain if it has no zero-divisors. Alternatively... Click here to read more

An integral domain is said to be a Euclidean ring if for every in there is defined a non-negative integer,... Click here to read more

A commutative ring with unity is called a field if its non-zero elements possesses a multiple inverse. Thus a ring... Click here to read more

Before giving the formal definition of an abstract vector space, we define what is known as an external composition in... Click here to read more

Theorem 1: The multiplicative inverse of a non-zero element of a field is unique. Proof: Let there be two multiplicative... Click here to read more

Let be a vector space over the field . Then a non-empty subset of is called a vector space of... Click here to read more

Linear Dependence Let be a vector space and let be a finite subset of . Then is said to be... Click here to read more

A subset of a vector space is said to be a basis of , if (i) consists of a linearly... Click here to read more