ADVANCED ALGEBRA UNIT-4: EUCLIDEAN RING

Dr. SHIVANGI UPADHYAY

ACADEMIC CONSULTANT UTTRAKHAND OPEN UNIVERSITY HALDWANI(UTTRAKHAND) [email protected]

Dr. Shivangi Upadhyay Advanced Algebra 1 / 20 Overview

1 Divisibility in a ring Divisor Great Common Divisor Unit Associates Prime element Theorems

2 Euclidean ring Definition Example Theorems

3 Unique factorization domain

Dr. Shivangi Upadhyay Advanced Algebra 2 / 20 Divisibility in a ring Divisibility in a ring

Divisor Let R be a commutative ring. A non-zero element a of R is called divisor of an element b ∈ R if there exist c ∈ R such that b = ac.

Example Let (Z, +,.) be a ring of integers, then 2 is divisor of 10 ∵ we can write 10 = 2.5, 5 ∈ Z In ring (Q, +,.) of rational numbers 2 divide 11 because 11/2 ∈ Q such that 11 = 2.11/2

Dr. Shivangi Upadhyay Advanced Algebra 3 / 20 Divisibility in a ring Great Common Divisor

Great Common Divisor Let R be a commutative ring. Let a, b ∈ R, then an element c ∈ R is called greatest common divisor (g.c.d) of a and b if c is a common divisor of a and b and if any other element x divides both a and b then x must divide c.

Examples In a ring (Z, +,.), 3 is a g.c.d of 9 and 15 and 5 is a g.c.d of 10 and 15.

Dr. Shivangi Upadhyay Advanced Algebra 4 / 20 Divisibility in a ring Unit

Unit Let R be a commutative ring with unity element 1. An element x ∈ R is called unit in R if it has multiplicative inverse in R i.e. if there exists y ∈ R such that xy = 1.

Examples In a field every non zero element is a unit, so in (R, +,.) and (Q, +,.) every non zero element is a unit. In (Z, +,.), 1 and -1 are the only units.

Dr. Shivangi Upadhyay Advanced Algebra 5 / 20 Divisibility in a ring Associates

Associates Let R be an integral domain. Two elements x and y of R are said to be associates if x divides y and y divides x.

Examples In (Z, +,.), 2 is an associates of 2 and -2.

Dr. Shivangi Upadhyay Advanced Algebra 6 / 20 Divisibility in a ring Prime element

Prime element Let R be an integral domain. A non zero, non unit element p of R is said to be prime or irreducible iff divisorof p are either units or its associates. i.e if p = xy where x, y ∈ R, then either x is a unit or y is a unit in R. Examples: In (Z, +,.), ±2, ±3, ±5, ..... are prime elements

Composite and Relatively prime Composite: A non zero element is said to be composite or reducible if it is neither a unit nor a prime. Example: In (Z, +,.), ±4, ±6, ±8, ..... are prime elements Relatively prime: If any two element of R have 1 as their g.c.d, then they are called relatively prime. Example:4 and 9 are relatively prime in Z.

Dr. Shivangi Upadhyay Advanced Algebra 7 / 20 Divisibility in a ring Prime element Theorems

Theorem 1 Let D be an integral domain. Let x and y be two non-zero elements of D. Let x and y be two non-zero elements of d, then x and y are asoociates iff x = ay, where a is a unit element in D.

Proof First suppose thet x and y are associates in an integral domain D. Then by definition of associates, x divides y and y divides x. Now

x|y =⇒ ∃ b ∈ D such that y = bx...... (1)

y|x =⇒ ∃ c ∈ D such that x = cy...... (2) We have to show that c is a unit in D.

Dr. Shivangi Upadhyay Advanced Algebra 8 / 20 Divisibility in a ring Prime element

Now, x = cy =⇒ x.1 = c(bx) =⇒ x.1 = (cb)x =⇒ x(1 − cb) = 0 =⇒ (1 − cb) = 0

(∵ x 6= 0 and D is an integral domain so it is without zero divisors) =⇒ cb = 1 =⇒ b and c both are units in D

Dr. Shivangi Upadhyay Advanced Algebra 9 / 20 Divisibility in a ring Prime element

Converse: Let x = ay, where a is unit in D, i.e., a−1 exists in D.

Now x = ay =⇒ y|x...... (3) Again x = ay =⇒ a−1x = a−1(ay) =⇒ a−1x = (a−1a)y =⇒ a−1x = y =⇒ x|y...... (4) By (3) and (4), x and y are associates.

Dr. Shivangi Upadhyay Advanced Algebra 10 / 20 Euclidean ring Euclidean ring

Definition A commutative ring R without zero divisors is said to be Euclidean ring if for every a 6= 0 ∈ R there is defined a on-negative integer d(a) such that (i) for all a, b ∈ R with b 6= 0, there exist q and r in R such that a = qb + r, where either r = 0 or d(r) < d(b) (Division Algorithm) (ii) for all a 6= 0, b 6= 0 ∈ R

d(a) ≤ d(b)

Dr. Shivangi Upadhyay Advanced Algebra 11 / 20 Euclidean ring

Example Ring (Z, +,.) of integers is a Euclidean ring for the Euclidean valuation d, defined by d(a) = |a|, ∀a 6= 0 ∈ Z Let a, b be two non zero elements in Z, then

d(ab) = |ab|

= |a||b|

≥ |a| = d(a)(∵ |b| ≥ 1) Hence, d(b) ≥ d(a)

Let a ∈ Z, b 6= 0 ∈ Z, then by divising algorithm in Z, there exist q and r in Z such that a = qb + r where 0 ≤ r < |b|, that is either r = 0 or d(r) < d(b). Also, Z is commutative ring with zero divisors Consequently Z is a Euclidean ring. Dr. Shivangi Upadhyay Advanced Algebra 12 / 20 Euclidean ring Theorems

Theorem 2 Every Euclidean ring is a principle ideal domain.

Proof Let R be a Euclidean ring. Then it is commutative and without zero divisors. In order to show that it is an integral domain, we have to show that it has unity element. We know that R is an ideal to itself. Also we know that Every Ideal I in a Euclidean ring R is of the form aR for some a ∈ I.

Dr. Shivangi Upadhyay Advanced Algebra 13 / 20 Euclidean ring Theorems

Hence, there exist a ∈ I such that R = aR, i.e.,

R = {ar|r ∈ R}

Thus, every element of R can be expressed as some multiple of a. Since a ∈ R so a also can be written as multiple of itself.

So, a = ab, for some b ∈ R...... (1)

Now, let x be any element of R then

x = ac, for some c ∈ R

=⇒ bx = b(ac) =⇒ bx = (ba)c (by associativity) =⇒ bx = (ab)c (by commutativity)

Dr. Shivangi Upadhyay Advanced Algebra 14 / 20 Euclidean ring Theorems

=⇒ bx = ac (by (1)) =⇒ bx = x (by (2)) Thus, bx = x = xb, ∀x ∈ R So, b is the unity element in R. Hence R is an integral-domain i.e., Every Euclidean ring is an integral domain As we know Every Ideal of R is Principle ideal Consequently R is a principle ideal domain.

Dr. Shivangi Upadhyay Advanced Algebra 15 / 20 Euclidean ring Theorems

Theorem 3 Let R be a Euclidean ring. Let x, y, z ∈ R such that x and y are relatively prime and x divides yz, then x divides z.

Proof Since x and y are relatively prime, therefore 1 is the greatest common divisor of x and y. As we know that if a and b any two non zero elements in Euclidean domain R then a and b have greatest common divisor such that c = (ma + nb), for some m, n ∈ R.

Dr. Shivangi Upadhyay Advanced Algebra 16 / 20 Euclidean ring Theorems

Therefore,1 can be expressed as

1 = mx + ny for some m, n ∈ R =⇒ 1.z = (mx + ny).z =⇒ z = mxz + nyz...... (1) Now, given that x|yz =⇒ x|nyz...... (2) also, x|x =⇒ x|mxz...... (3) By (2) and (3), we have

x|(mxz + nyz) =⇒ x|z (by(1))

Dr. Shivangi Upadhyay Advanced Algebra 17 / 20 Unique factorization domain

Definition Let R be an integral domain. Then R is said to be a unique factorization domain(UFD) if any non-zero element of R is either a unit or it can be expressed as the product of a finite number of prime elements and this product is unique up to associates. Thus, if a ∈ R is a non-zero, non-unit element, then (i) a = x1x2...... xm, xi (1 ≤ i ≤ m) are prime in R. (ii) If a = y1y2.....yn, also, where yj (1 ≤ i ≤ n) are prime in R, then m = n and each xi is an associate of some xi and vice-versa.

Dr. Shivangi Upadhyay Advanced Algebra 18 / 20 Unique factorization domain

Example Every field F is Unique factorization domain since every non zero element is invertible with respect to multiplication, i.e. every non zero element in F is necessarily a unit. Ring Z and Z(i)

Dr. Shivangi Upadhyay Advanced Algebra 19 / 20 Unique factorization domain

THANK YOU

Dr. Shivangi Upadhyay Advanced Algebra 20 / 20