Ideals, quotient rings, homomorphisms

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IDEALS, QUOTIENT RINGS, HOMOMORPHISMS

The theory of ideals is closely connected with the concepts of congruence, congruence classes, and residue classes of integers mod n.

Congruence. Let n be a positive integer. We say two integers are congruent “modulo n" if they differ by a multiple of the integer n. For example, if n = 5 we can say that 3 is congruent to 23 modulo 5 (and write it as 3 ≡ 23 mod 5) since the integers 3 and 23 differ by 4x5 = 20. The statement a ≡ b (mod n) is equivalent to the statements “a - b is a multiple of n” or “a - b is divisible by n”. The integer n is called the modulus of the congruence.

An alternate definition: a ≡ b (mod n) if and only if a and b have the same remainder when divided by n.

A similar idea to that of modulus arithmetic occurs in everyday life in telling time. Clocks go up to12 and then start over.

The congruence relation a ≡ b (mod n) creates a set of equivalence classes on the set of integers in which two integers are in the same class if they are congruent modulus n, i.e. if they leave the same remainder when divided by n.

Syn. Modulus, modulo, mod

Residue classes of integers mod n. The congruence relation a b (mod n) on the set of integers I separates the integers into n equivalence classes,

[0]_n, [1]_n, [2]_n, ... ,[n-1]_n,

called residue classes modulo n. Each equivalence class [r]_n consists of all integers congruent to r where r is one of the integers 0,1,2, ... ,n-1. These n integers 0,1,2, ... ,n-1 are called the class representatives. Thus equivalence class [3]_n consists of all integers congruent to 3 mod n where the integer 3 is the class representative.

Example. The residue classes of integers mod 4 are:

[0]₄ = { ... , -16, -12, -8, -4, 0, 4, 8, 12, 16, ... }

[1]₄ = { ... , -15, -11, -7, -3, 1, 5, 9, 13, 17, ... }

[2]₄ = { ... , -14, -10, -6, -2, 2, 6, 10,14, 18, ... }

[3]₄ = { ... , -13, -9, -5, -1, 3, 7, 11, 15, 19, ... }

Set of all residue classes mod n, I/(n). We denote the set of all residue classes modulo m by I/(n). For example,

I/(4) = { [0]₄, [1]₄, [2]_4,[3]₄ }

and

I/(n) = { [0]_n, [1]_n, [2]_n, ... ,[n-1]_n }

Note that I/(n) consists of a set of sets.

Modular arithmetic (or arithmetic modulo n). A modular arithmetic or arithmetic modulo n is obtained by using only the class representatives 0, 1, 2, ... , n-1 and defining addition and multiplication by letting the sum a + b and the product ab be the remainder after division by n of the ordinary sum and product of a and b. E.g. if n = 7, then 2 + 5 0, 3∙6 4, and the multiplicative inverse of 2 is 4, since 2∙4 ≡1. If n = 15, then 3 has no multiplicative inverse, since a multiplicative inverse a would have the property that 3∙a - 1 = k∙15 for some integer k and that 3(a - 5k) = 1. Arithmetic modulo n is a commutative ring with unit element; if n is a prime, then arithmetic modulo n is a field.

James & James. Mathematics Dictionary. Congruence.

Using modular arithmetic residue classes mod n can be added and multiplied and form a ring.

Ideal.

Def. Left ideal. An additive subgroup H of a ring R for which rx ∈ H for all r ∈ R and x ∈ H.

Def. Right ideal. An additive subgroup H of a ring R for which xr ∈ H for all r ∈ R and x ∈ H.

An additive subgroup of a ring R is called a two-sided ideal (or simply an ideal) if it is both a left and right ideal.

Criterion for a subset of a ring to be an additive group. A subset I of a ring R is an additive group if and only if a - b belongs to I whenever a and b belong to I. This fact is a consequence of the following theorem on groups:

Theorem. A nonempty subset H of a group G is a subgroup of G if and only if b^-1a is in H for all a,b in H.

When the group operation is addition, b^-1a becomes a - b.

A subset I of a ring R is an ideal if and only if I is non-empty and a - b, ax and xa are in I for all

a,b ∈ S and all x ∈ R.

An ideal (left ideal, right ideal or two-sided ideal) is a subgroup of the additive group of a ring. Since the additive group of a ring is Abelian, all of its subgroups are invariant subgroups (normal subgroups). Thus any ideal of a ring is necessarily a normal subgroup with respect to the operation of addition. Any subring of a ring is necessarily normal with regard to addition.

Proper and improper ideals. Every ring has at least two ideals: 1) The null ideal consisting of the zero element alone, and 2) R itself. These two are called improper ideals. All other ideals are proper ideals.

Simple ring. A ring having no proper ideals.

Prime ideal. An ideal I in a commutative ring R is said to be a prime ideal if, for arbitrary elements r,s of R, the fact that r∙s ∈ I implies either r ∈ I or s ∈ I.

Example. In the ring I the ideal K = {14r: r ∈ I} is not a prime ideal since, for example,

28 = 4∙7 ∈ K but neither 4 nor 7 is in K. On the other hand, the ideal K = {7r: r ∈ I} is a prime ideal.

Theorem. In the ring I a proper ideal I = {mr: r ∈ I, m 0} is a prime ideal if and only if m is a prime integer.

Coset of an ideal. By the term coset of an ideal or subring of a ring is meant the coset with respect to the operation of addition. Thus if H is a subring or ideal, a left coset aH is written a + H to make that fact clear. The coset a + I of an ideal I is called a residue class of I in the ring R. The cosets of an ideal I of a ring R partition R into a set of equivalence classes, or residue classes. Two elements a and b of R are called “congruent modulo I” [written a ≡ b (mod I) ] if they fall into the same residue class and they fall into the same modulo class if and only if a - b ∈ I.

The rule for adding cosets is defined (the usual rule for adding complexes). It is possible to define multiplication on cosets in such a way that the cosets (i.e. residue classes) form a ring called the quotient ring R/I.

Operations on residue classes.

Sum of two residue classes. The sum of two residue classes I + r and I + s is given by

(I + r) + (I + s) = I + (r + s).

This rule corresponds to the usual rule for the sum of cosets.

Def. Product of two residue classes. The product of two residue classes I + r and I + s is given by

(I + r) x (I + s) = I + (rs).

by definition.

Quotient ring (R/I). The quotient ring R/I is the set of all cosets of I i.e. all sets a + I for all a ∈ R. The addition and multiplication operations are those defined for cosets. The zero element of R/I is I.

Principal ideal. An ideal is a principal ideal if it contains some element “a” such that all elements of the ideal are multiples of this element a. The element a is called the generating element. A principal ideal generated by an element a is denoted by (a). In other words, a principal ideal is an ideal that is an additive cyclic group generated by some element a.

Example. The set

{ .... -12, -9, -6, -3, 0, 3, 6, 9, 12, ....}

with addition as the group operation in the ring of integers I is a principal ideal in I. It is generated by the element 3 and is denoted as the principal ideal (3). It is a cyclic group with an infinite number of members. In the same way principal ideals corresponding to (4), (5), ... , (n) represent similar sets. Any integer will generate a principal ideal in I.

Connection between quotient rings and congruence classes of integers. The cosets of the principal ideal (n) of the ring I of all integers partition I into n residue classes where two integers a and b fall into the same residue class if a ≡ b (mod n) i.e. if a - b is a multiple of n. This set of n residue classes corresponds to the quotient group I/(n).

Example. The principal ideal (4) of the ring of integers I consists of the set

{ .... , -20, -16, -12, -8, -4, 0, 4, 8, 12, 16, 20, ... }

The cosets of (4) consist of a + (4) for all a ∈ I.

The residue classes of integers mod 4 are:

[0]₄ = { ... , -16, -12, -8, -4, 0, 4, 8, 12, 16, ... }

[1]₄ = { ... , -15, -11, -7, -3, 1, 5, 9, 13, 17, ... }

[2]₄ = { ... , -14, -10, -6, -2, 2, 6, 10,14, 18, ... }

[3]₄ = { ... , -13, -9, -5, -1, 3, 7, 11, 15, 19, ... }

These four residue classes correspond to the four cosets of the principal ideal (4) and thus to the four members of the quotient group I/(4). Every integer in I falls into one of these four residue classes. The four classes partition the set of integers I into four disjoint equivalence classes. The residue class [0]₄ corresponds to the principal ideal (4) itself.

Thus we see that the general concept of “residue classes modulo some ideal I” reduces to the concept of “residue classes modulo some integer” for the case of a principal ideal (n) in the ring of integers.

Principal ideal ring. A principal ideal ring is a commutative ring in which every ideal is a principal ideal.

Example. The ring I (set of all integers) is a principal ideal ring.

Theorems.

1] The intersection of an arbitrary system of ideals is again an ideal.

2] The intersection of all ideals of a ring R which contain a fixed element “a” of R is an ideal. It is a principal ideal generated by element a.

3] In the ring I of integers the subgroup P of all integral multiples of any integer p is an ideal in I.

4] In the ring I of integers every ideal is a principal ideal.

Homomorphisms between rings. A homomorphism that maps a ring R onto a ring R' defines a partition of R: a class R_a is formed by all elements a in R mapping into the same image a' in R'. The class I of R which maps into the zero element of R' is an ideal in R and the other classes R₁, R₂, R₃, ... of R mapping onto single images in R' are cosets (or residue classes) of this ideal I. The ideal I along with its cosets form a ring (under the rules for adding and multiplying cosets) called the quotient ring R/I.

Kernel of a homomorphism. If a homomorphism maps a ring R onto a ring R*, then the kernel of the homomorphism is the set I of elements which map onto the zero element of R*. The kernel I is an ideal and R* is isomorphic with the quotient ring R/I.

References.

A.D. Aleksandrov, A.N. Kolmogorov, M.A. Lavrentev, editors. Mathematics, its Content, Methods and Meaning. Volume III.

James / James. Mathematics Dictionary.

Frank Ayres. Modern Algebra. (Schaum)

Joong Fang. Abstract Algebra. (Schaum)

Garrett Birkhoff, Saunders Mac Lane. A Survey of Modern Algebra.

R. A. Beaumont, R. W. Ball. Introduction to Modern Algebra and Matrix Theory.

B. L. Van der Waerden. Modern Algebra.