Introduction to Rings in Algebra
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
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