Rings, Ideals, Quotient Rings - Ring Theory, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

A ring A is a set with + , • such that

In this course all rings A are commutative, that is, and have an identity element 1 (easily seen to be unique)

Multiplication will be denoted by juxtaposition, and simple facts used without comment, such as

Call a subset S of a ring A a subring if

Condition (ii) is easily seen to be equivalent to

Examples:

(1) Z is the only subring of Z .

(2) Z is a subring of Q , which is a subring of R , which is a subring of C .

(3) Z[i] = { a + bi | a, b ∈ Z }

the ring of Gaussian integers is a subring of C 

(4) Z_n = { 0, 1, . . . , n − 1}

with addition and multiplication mod n .

(Alternatively Z_n may be defined to be the quotient ring Z/nZ , defined below)

(5) R any ring, x an indeterminate. Put

R[[x]] = {a₀ + a₁x + a₂x² + . . . | a₀, a₁, . . . ∈ R} , the set of formal power series over R , which becomes a ring under addition and multiplication of power series. Important subring:

R[x] = {a₀ + a₁x + . . . +a_nxⁿ | n ≥ 0 , a₀, a₁, . . . , a_n ∈ R} 

the ring of polynomials over R .

Call a mapping f : A → B (where A and B are rings) a ring homomorphism if

in which case the following are easily checked:

An isomorphism is a bijective homomorphism, say f : A → B , in which case we write

A ≌ B or f : A ≌ B.

It is easy to check that

A nonempty subset I of a ring A is called an ideal, written

In particular I is an additive subgroup of A , so we can form the quotient group the group of cosets of I ,

A/I = { I + a | a ∈ A} ,

with addition defined by, for a, b ∈ A , (I + a) + (I + b) = I + (a + b) .

Further A/I forms a ring by defining, for a, b ∈ A ,

(I + a) (I + b) = I + (a b) .

— only tricky bit is first checking multiplication is well-defined:

If I + a = I + a′ and I + b = I + b′ then

a − a′ , b − b′ ∈ I ,

so

ab − a′b′ = ab − ab′ + ab′ − a′b′

= a(b − b′) + (a − a′)b′ ∈ I 

yielding I + ab = I + a′b′ .

We call A/I a quotient ring.

The mapping

φ : A → A/I , x → I + x

is clearly a surjective ring homomorphism, called the natural map, whose kernel is

ker φ = {x ∈ A | I + x = I} = I .

Thus all ideals are kernels of ring homomorphisms.

The converse is easy to check, so

The following important result is easy to verify:

Proposition: and φ : A → A/I be the natural map. Then (i) ideals J of A/I have the form J = J/I = {I + j | j ∈ J} for some J such that I ⊆ J A ; (ii) φ−1 is an inclusion-preserving bijection between ideals of A/I and ideals of A containing I .

Example: The ring Z_n = {0, 1, . . . , n − 1} with mod n arithmetic is isomorphic to Z/nZ : follows from the Fundamental Homomorphism

Theorem, by observing that the mapping f : Z → Z_n where

f (z) = remainder after dividing z by n is a ring homomorphism with image Z_n and kernel nZ.