GATE EXAM : RING THEORY NOTES (GATE 2023)

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Table of Content Ring Structure &  Properties Unity Unit Zero Divisor Integral Domain Characteristic of a ring Subring Ring Homomorphism Irreducibility tests Principal Ideal Domain Euclidean Domain Unique Factoriation Domain (UFD)

Must Read: GATE 2023 Brochure

Ring Structure & Properties   

A non-empty set R, together with two binary composition + and is said to form a ring if the following axioms are satisfied :

(i) a + b ∈ R and a.b ∈ R whenever a,b ∈ R.

(ii) (a + b) + c = a + (b + c) for all a, b, c ∈ R

(iii) a + b = b + a for a, b ∈ R

(iv) ∃ some element 0 (called zero) in R s.t.  a + 0 = 0 + a = a  for all a ∈ R

(v) For each a ∈ R, ∃ an element (-a) ∈ R s.t. a + (-a) = 0 = (-a) + a

(vi) a . (b . c) = (a . b) . c  for all a, b, c ∈ R

(vii) a . (b + c) = a.b + a.c

       b + c) . a = b.a + c.a  for all a, b, c ∈ R

       i.e. (R, +) is abelian group and (R, .) is semi group.

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Unity 

Multiplication identity in any ring is called unity. If ∃ e ∈ R with a ∈ R, a.e = e.a = a then e is said to be as unity of ring and it is denoted by 1.

Unit 

Let (R, +, .) be any ring and a Î R then a is said to be unit element of R, if ∃ b ∈ R a.b = b.a = 1

Zero divisor 

Let R be a ring and 0 ≠ a ∈ R then a is said to be zero divisor if ∃  0 ≠ b ∈ R s.t.a.b = 0 = b.a

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Integral Domain 

A commutative ring with unity without zero divisors is called an integral domain.

Characteristic of a ring 

The smallest positive integer n is said to be characteristic of a ring R

If na = 0  for all a ∈ R i.e. if a + a + a ….. + a = 0   (n-times)

If no such positive integer exist then characteristic of R is 0 and it is denoted by 0.

Subring

Let (R, +, .) be a ring.

Then a non-empty subset S of R is called a subring of R if (S, +,.) is a ring itself.

SubringTest:

Let S≠Φ and S⊆R,

then S is subring of R if and only if

(i)        ab ∈ S a, b ∈ S

(ii)       a – b ∈ S for all a, b ∈ S

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Ideal 

• Left Ideal : Let R be a ring and I be a subring of R, then I is said to be left ideal of R if for each r ∈ R and a ∈ I we have ra ∈ I.
• Right Ideal : Let R be a ring and I be a subring of R, then I is said to be right ideal of R.

            If for each a ∈ I, r ∈ R we have ar ∈ I.

• Ideal : A subring I of a ring R is said to be ideal. If I is both left ideal and right ideal.

• IdealTest: A non- empty subset A of a ring R is an ideal of R if

            (i) a – b ∈ A whenever a,b ∈ A.

            (ii) ra&ar ∈ A whenever, a ∈A & r ∈ R.

Maximal Ideal

An ideal M in a ring R is called maximal ideal of R if M ≠ R and if ∃ ideal U of R such that M⊆U⊆R. Then either M=U or U = R.

Prime Ideal :

Let R be a commutative ring. An ideal P is called a prime ideal if P≠R and whenever ab ∈ P then either a ∈ P or b ∈ P.

Results:

• pZ x Z = {(px, y) | x,y ∈ Z} is maximal ideal and prime ideal of ZxZ where p is prime.
• {(0, 0),(0,x 1)} and {(0, 0), (1, 0)} are the maximal ideals and prime ideals of Z₂ x Z₂.

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Ring Homomorphism 

Let (R, +, .) & (R’, ⊕, Θ) two rings then a map :R → R’ is defined as ring homomorphism if

(i) Φ(a+b) = Φ(a) ⊕ Φ(b); for all a,b ∈ R

(ii) Φ(a.b) = Φ(a) Θ Φ(b); for all a,b ∈ R

Kernel of Homomorphism:

• Let Φ be a homomorphism from a ring R to a ring R’.
• Then ker Φ is defined as: ker Φ = {r ∈ R | Φ(r) = 0}

Properties :

• kerΦ = {0} if and only if Φ is one – one .
• kerΦ is an ideal in R.

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Ring of Polynomials

Let R be a ring, then the set R[x]={a_nxⁿ+ a_n-1x^n-1….+a₁x+a₀: a_i ∈ R, 
n ∈ N U {0}} forms a ring with respect to addition and multiplication of polynomials.

Degree of a Polynomial :

• Let f(x) = a₀ + a₁x + …… + a_nxⁿ ∈ R[x], we say that f(x) is a non-zero polynomial if atleast one of the coefficients a₀,a₁, ….,a_n. is non-zero, then we say that f(x) has degree nifa_n≠0.
• We write it as deg f(x) = n or deg (f) = n.

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Irreducibility Tests

(i) If a polynomial f(x) is of degree >1 and f(a)= 0 for some a ∈ F. Then f(x) is reducible over F where F is a field

Reducibility Test for degree 2 and 3:
Let F be a field if f(x) ∈ F[x]and degf(x)=2 or 3. then f(x) is reducible over F if and only if f(x) has a zero in F.

Let p be a prime and suppose that f(x) ∈ Z[x] with degf ≥ 1.

Let f(x) be the polynomial in Z_p[x] obtained from f(x) by reducing all the coefficientsof f(x) modulopiff(x)is irreducible over Z_p
and degf(x)=degf_p(x)then f(x) is irreducible over Q.

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Eisenstein’s Criterion:
Let f(x)=a_nxⁿ + a_n-1x^n-1 + ….. + a₀ ∈ Z[x]. 

If ∃a prime p such that p|a₀.p|a₂,….p|a_n-1and p²∤ a₀, p ∤ a_n, then f(x) is irreducible over Q.

Example:- f(x) = 3 + 6x + 12x² + x³ ∈ Z[x]  is irreducible over Q.

Sol.      p = 3

            Here 3 | 3, 3 | 6, 3 | 12 but 3 ∤ 1 and 3²∤ 3

            Then f(x) is irreducible over Q.

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Principal Ideal Domain 

An integral domain R is said to be principal ideal domain if each ideal I of R is P.I.

i.e.       I = <a> = {ar | r ∈ R}

(i) every field is P.I.D. but conversely need not be true.

Example : Q[x]

            (ii) Z_p[x] is P.I.D. if p-prime.

            (iii) Z[x] is not P.I.D. (∃, I =<2, x> which is not PI

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Euclidean Domain

Let R be an integral domain if ∃ a norm on R s.t.

(i)        N(a.b) ≥ N(a)  for all a, b ≠ 0 ∈ R

(ii)       For any a, 0≠b∈R  ∃ q, r ∈Rs.t.a = bq + r, r = 0 or N(r) <N(b) then R is said to be Euclidean domain.

Example:-       (i)        Every field is E.D.

                        (ii)       Z, Z[i] & Z[√2] are E.D.

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Unique Factoriation Domain (UFD) 

An I.D. is called UFD if it satisfy the properties.

• Each non-zero element of R is either unit or can be expressed as a product of finite number of irreducible element of R.

Example :-      (i)        Z[x] is UFD.

            (ii)       Every field is UFD.

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