Quotient module
Encyclopedia
In abstract algebra
, given a module
and a submodule, one can construct their quotient module. This construction, described below, is analogous to how one obtains the ring
of integer
s modulo an integer n, see modular arithmetic
. It is the same construction used for quotient group
s and quotient ring
s.
Given a module A over a ring R, and a submodule B of A, the quotient space
A/B is defined by the equivalence relation
for any a and b in A. The elements of A/B are the equivalence classes [a] = { a + b : b in B }.
The addition
operation on A/B is defined for two equivalence classes as the equivalence class of the sum of two representatives from these classes; and in the same way for multiplication by elements of R. In this way A/B becomes itself a module over R, called the quotient module. In symbols, [a] + [b] = [a+b], and r·[a] = [r·a], for all a,b in A and r in R.
s, and the R-module A = R[X], that is the polynomial ring
with real coefficients. Consider the submodule
of A, that is, the submodule of all polynomials divisible by X2+1. It follows that the equivalence relation determined by this module will be
Therefore, in the quotient module A/B one will have X2 + 1 be the same as 0, and such, one can view A/B as obtained from R[X] by setting X2 + 1 = 0. It is clear that this quotient module will be isomorphic to the complex number
s, viewed as a module over the real numbers R.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, given a module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
and a submodule, one can construct their quotient module. This construction, described below, is analogous to how one obtains the ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
of integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s modulo an integer n, see modular arithmetic
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
. It is the same construction used for quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
s and quotient ring
Quotient ring
In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
s.
Given a module A over a ring R, and a submodule B of A, the quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
A/B is defined by the equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
- a ~ b if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
b − a is in B,
for any a and b in A. The elements of A/B are the equivalence classes [a] = { a + b : b in B }.
The addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
operation on A/B is defined for two equivalence classes as the equivalence class of the sum of two representatives from these classes; and in the same way for multiplication by elements of R. In this way A/B becomes itself a module over R, called the quotient module. In symbols, [a] + [b] = [a+b], and r·[a] = [r·a], for all a,b in A and r in R.
Examples
Consider the ring R of real numberReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, and the R-module A = R[X], that is the polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
with real coefficients. Consider the submodule
- B = (X2 + 1) R[X]
of A, that is, the submodule of all polynomials divisible by X2+1. It follows that the equivalence relation determined by this module will be
- P(X) ~ Q(X) if and only if P(X) and Q(X) give the same remainder when divided by X2 + 1.
Therefore, in the quotient module A/B one will have X2 + 1 be the same as 0, and such, one can view A/B as obtained from R[X] by setting X2 + 1 = 0. It is clear that this quotient module will be isomorphic to the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s, viewed as a module over the real numbers R.