Rings Field and Vector Space

  • Introduction to Rings in Algebra

    The concept of a group has its origin in the set of mappings or permutations, of a set onto itself. So far we have considered sets with one binary operation only. But rings are the motivation which arises from the fact that integers follow a definite pattern with respect to the addition and multiplication. Thus […]

  • Elementary Properties of Ring

    Some basic elementary properties of a ring can be illustrated with help of following theorems and these properties are used in developing further concepts in rings and these properties are building of rings. Theorem: If is a ring, then for all are in . (a) (b) (c) Proof: (a) We know that Since is a […]

  • Examples of Ring

    Example 1: A Gaussian integer is a complex number , where and are integers. Show that the set of Gaussian integers forms a ring under ordinary addition and multiplication of complex numbers.     Solution: Let and be any two elements of then and These are Gaussian integers and therefore is closed under addition as well […]

  • Special Types of Rings

    1. Commutative Rings: A ring is said to be a commutative, if the multiplication composition in is commutative. i.e. 2. Rings with Unit Element: A ring is said to be a ring with unit element if has a multiplicative identity, i.e. if there exist an element denoted by , such that The ring of all […]

  • Cancellation Laws in a Ring

    Cancellation Laws in a Ring: We say that cancellation laws hold in a ring , if and where are in . Thus in a ring with zero divisors, it is impossible to define a cancellation law. Theorem: A ring has no divisor of zero if and only if the cancellation laws holds in . Proof: […]

  • Subrings

    Let be a ring. A non –empty subset of the set is said to be a subring of if is closed under addition and multiplication in and itself is a ring for those operations. If is any ring, then and are always subrings of . These are said to be improper subrings. The subrings of […]

  • Intersection of Subrings

    Theorem: The intersection of two subrings is a subring. Proof: Let and be two subrings of ring . Since and at least . Therefore is non-empty. Let , then and and and But and are subrings of , therefore and and and Consequently, and . Hence, is a subring of .

  • Ideals in Ring

    Ideals: Let be any ring and a subring of , then is said to be right ideal of if and left ideal of if . Thus a non-empty subset of , is said to be a ideal of if: (i) is a subgroup of under addition. (ii) For all and , both and . Proper […]

  • Integral Domain in Ring

    Integral Domain: A commutative ring with unity is said to be an integral domain if it has no zero-divisors. Alternatively a commutative ring with unity is called an integral domain if for all , . Examples: (i) The set of integers under usual addition and multiplication is an integral domain as for any two integers, […]

  • Euclidean Ring

    An integral domain is said to be a Euclidean ring if for every in there is defined a non-negative integer, to be denoted by , such that: (i) For all , both non-zero, , (ii) For any , both non-zero, there exist such that when either or . Note: The set of integer that depends […]

  • Field in Algebra

    A commutative ring with unity is called a field if its every non-zero elements possesses a multiple inverse. Thus a ring in which the elements of different from form an abelian group under multiplication is a field. Hence, a set , having at least two distinct elements together with two operations and is said to […]

  • Vector Space

    Before giving the formal definition of an abstract vector space we define what is known as an external composition in one set over another. We have already define a binary composition in a set as a mapping of to . This may be referred to as an internal composition in . Let now and be […]