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 $$a + ib$$, where $$a$$ and $$b$$ are integers. Show that… Click here to read more

1. Commutative Rings A ring $$R$$ is said to be a commutative if the multiplication composition in $$R$$ is commutative,… Click here to read more

Cancellation Laws in a Ring We say that cancellation laws hold in a ring $$R$$ if $$ab = bc\,\left( {a… Click here to read more

Let $$R$$ be a ring. A non–empty subset $$S$$ of the set $$R$$ is said to be a subring of… Click here to read more

Theorem: The intersection of two subrings is a subring. Proof: Let $${S_1}$$ and $${S_2}$$ be two subrings of ring $$R$$…. Click here to read more

Ideals: Let $$\left( {R, + , \times } \right)$$ be any ring and $$S$$ be a subring of $$R$$, then… 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 $$R$$ is said to be a Euclidean ring if for every $$a \ne 0$$ in $$R$$ there… 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 $$V$$ be a vector space over the field $$F$$. Then a non-empty subset $$W$$ of $$V$$ is called a… Click here to read more

Linear Dependence Let $$V\left( F \right)$$ be a vector space and let $$S = \left\{ {{u_1},{u_2}, \ldots ,{u_n}} \right\}$$ be… Click here to read more

A subset $$S$$ of a vector space $$V\left( F \right)$$ is said to be a basis of $$V\left( F \right)$$,… Click here to read more