Chapter 2 Subrings and Ideals

2.1 INTRODUCTION

In this chapter, we discuss, subrings, sub fields. Ideals and quotient ring. We begin our study by defining a subring. If (R, +, ×) is a ring and S is a non-empty subset of R, then ‘+’ and ‘×’, may induce binary operations ‘+’ and ‘×’ respectively on S. If S is a ring with respect to these induced operations, then we call S, a subring of R.

2.2 SUBRING

Definition 2.1 Let (R, +, ×) be a ring and let S be a non-empty subset of R. If (S, +, ×) is a ring, then S is called a subring of R. Every non-zero ring R has two trivial subrings, viz. the ring itself and the zero ring consisting of the zero element of the ring R. {0} and R are called the improper subrings of R. If S is a subring of R, then (i) S is a subgroup of additive group R. i.e., (S, +) is a subgroup of (R, +). (ii) S is closed with respect to multiplication.

éùa 0 Example 1: The set S of all 2 ´ 2 matrices of the type êú where a, b, c are ëûbc ´ integers is subring of the ring M2 of all 2 matrices over Z. Example 2: The set of integers Z is a subring of the ring of real numbers. Theorem 2.1: A non-empty subset S of a ring R is a subring of R if and only if a - b Î S and ab Î S for all a, b Î S Proof: Let S be a subring of R and let a, b Î S. Then S is a subgroup of R under addition. Hence, b Î S Þ - b Î S and, a Î S, b Î S Þ a Î S, - b Î S Þ a + (- b) Î S Þ a - b + S

Chapter-02 2nd Proof Anvi T12.p6539 10/5/10, 11:53 AM Subrings and Ideals 41 Hence, S is a subring of R.

Example 2: Show that S = {0, 3} is a subring of (z6, +6, x6) under the operations +6 and x6. Solution: We construct the composition tables as follows: + ´ 6 03 6 03 003 000 330 303 From the above composition tables Î Î Þ - - Î a S, b S a +6 ( b) = a b S ´ Î " Î a 6 b S a, b S Hence, S is a subring of R. Example 3: Let m be any fixed integer and let S be any subset of Z, the set of integers, such that S = {..., - 3m, - 2m, - m, 0, m, 2m, 3m, ...} show that S is a subring of (z, +, ×) Solution: Let rm, s m Î S, then r, s Î Z now rm - sm = (r - s)m Î Z (Q r - s Î Z " r, s Î Z)

and, (rm) (sm)= (rsm) m Î Z (Q r s m Î Z " r, s, m Î Z) \ S is a subring of (Z, +, ×) Theorem 2.2: The necessary and sufficient conditions for a non-empty subset S of a ring R to be a subring of R are: (i) S + (- S) = S and (ii) SS Í S Solution: Let S be a subring of R Then (S, +) is a subgroup of (R, +) let a + (- b) Î S + (- S), then a + (- b) Î S + (- S) Þ a Î S, - b Î - S Þ a Î S, b Î S Þ a - b Î S Þ a + (- b) Î S Thus, S + (- S) Í S (1) 0 Î S Þ 0 Î - S Now a Î S, 0 Î - S Þ a + 0 Î ST (- S) Hence, S Í S + (- S) (2)

Chapter-02 2nd Proof Anvi T12.p6541 10/5/10, 11:53 AM 42 Linear Algebra From (1) and (2), we have S + (- S)= S Again, S is a subring of R Þ S is closed under multiplication. Thus, a Î S, b Î S Þ a b Î S but ab Î SS Now ab Î SS Þ a Î S, b Î S Þ ab Î S

Þ SS Í S Hence, proved. Conversely, let S + (- S) = S and SS Í S " a, b Î S, we have ab Î SS Þ ab Î S (Q SS Í S) Again S + (- S)= S Þ S + (- S) Í S we have a + (- b) Î S + (–S) Í S Thus, Þ a + (- b) Î S " a, b Î S Þ a - b Î S " a, b Î S Thus, a - b Î S, ab Î S " a, b Î S Hence, S is a subring of R. Theorem 2.3: The intersection of two subrings of a ring R is a subring of R.

Proof: Let S1 and S2 be two subrings of a ring R. Î Î Î Ç 0 S1, 0 S2, therefore, 0 S1 S2 Ç ¹ f Thus, S1 S2 Î Ç Now let a, b S1 S2, then Î Ç Þ Î Î a S1 S2 a S1, a S2 Î Ç Þ Î Î b S1 S2 b S1, b S2

But S1, S2 are subrings of R, therefore, Î Î Þ - Î Î a S, b S1 a b S1, ab S1 Î Î Þ - Î Î and, a S2, b S2 a b S2, ab S2 - Î - Î Þ - Î Ç a b S1, a b S2 a b S1 S2 Î Î Þ Î Ç ab S1, ab S2 ab S1 S2 - Î Ç Î Ç " Î Ç Consequently, a b S1 S2, ab S1 S2 a, b S1 S2 Ç Hence, S1 S2 is a subring of R. È Theorem 2.4: Let R be a ring and S1, S2 be two subrings. Then S1 S2 is a Í Í subring of R if and only if S1 S2 or S2 1.

Chapter-02 2nd Proof Anvi T12.p6542 10/5/10, 11:53 AM 44 Linear Algebra 3. Prove that the z of integers is a subring of R, the set real numbers under addition and multiplication. 4. Show that the set of n ´ n matrices over the rational numbers is a subring of n ´ n matrices over the real numbers under addition and multiplication matrices. ´ ´ 5. Show that ({0, 2, 4}, +6, 6) is a subring of (Z6, +6, 6)

where, Z6 = {0, 1, 2, 3, 4, 5}. 6. R is an integral domain. Show that the set S = {mx : x Î R, m is a fixed integer} is a subring of R. 7. If R is a ring, then show that the set N(a)= {x Î R; ax = xa} is a subring of R.

2.3 IDEALS

2.3.1 Left Ideal Definition 2.2: A non-empty subset U of a ring R is called a left ideal if (i) a Î U, b Î U Þ a - b Î U (ii) a Î U, r Î R Þ r a Î U Example: In the ring M of 2 ´ 2 matrices over integers consider the set U ìüéùa 0 = íýêú:,abÎ Z îþïïëûb 0 éù00 êú Î U Þ U ¹ f ëû00 Let éùa 0 éùc 0 A = êú, B = êú Î U, then ëûb 0 ëûd 0

éùac- 0 A - B = êú Î U ëûbd- 0 Also éùab éùa 0 P = êú Î M, A = êú Î U ëûgd ëûb 0 éùéùaba 0 Þ PA = êúêú ëûëûgdb 0

Chapter-02 2nd Proof Anvi T12.p6544 10/5/10, 11:53 AM Subrings and Ideals 45

éùabab+ 0 = êú Î U ëûgdab+ 0 This shows that U is a left ideal of M 2.3.2 Right Ideal Definition 2.3: A non-empty set U of a ring R is called a right ideal of R if (i) a Î U, b Î U Þ a - b Î U, and (ii) a Î U, r Î R Þ ar Î U. Example: Let M be the ring of 2 ´ 2 matrices one integers. Consider ìüéùab U = íýêú:,abÎ z îþïïëû00 éù00 êú Î U Þ U ¹ f ëû00 éùab éùcd éùacbd-- A = êú Î U B = êú Î U Þ A - B = êú Î 0 ëû00 ëû00 ëû00

éùab éùa 0 and, P = êú, A = êú Î U ëûgd ëûb 0 éùéùabab we have AP = êúêú ëûëû00gd éùababag++ bd = êú Î U ëû00 Hence, U is a right ideal of M. 2.3.3 Ideal Definition 2.4: A non-empty set U of a ring R is called an ideal (two-sided ideal) of R if (i) a Î U, b Î U Þ a - b Î U, and (ii) a Î U, r Î R Þ a r Î U and r a Î U Example: The set E of even integers is an ideal of the ring Z of integers. a, b Î E Þ a = 2m, b = 2n for some integers m and n we have a - b = 2m - 2n = 2(m - n) Î E also r Î Z, a Î E Þ ra = r(2m) = 2(rm) Î E and, ar = (2m) r = 2(mr) Î E Hence, E is an ideal of Z.

Chapter-02 2nd Proof Anvi T12.p6545 10/5/10, 11:53 AM Subrings and Ideals 47 Consider a(xr) a(xr)= (ax) r = 0 × r = 0 Þ Î " Î Î x r S r R1 x S Thus, S is a right ideal of R. Example 3: If R is a commutative ring with unity, then the ideal Ra is the small- est ideal containing a. Proof: Let < a > = Ç {U: U is a ideal of R and a Î U} Clearly, < a > is the smallest ideal which contains a. We shall show that < a > = Ra. Since Ra is an ideal and a Î Ra, we have < a > Ì Ra Let U be any ideal of R such that a Î U For any r Î R, r a Î U by the definition of an ideal. Hence, Ra Ì U. Since U is an arbitrary ideal containing a it follows that R a Ì Ç {U: U is an ideal and a Î U} i.e., Ra Ì < a > Hence, Ra = < a > Theorem 2.6: If R is a ring with unity and U is an ideal of R such that 1 Î U, then U = R. Proof: U is an ideal of R Þ U Í R (1) Let x Î R Now x Î R, 1 Î U Þ n × 1 Î U (since U is a ideal of R) Þ x Î U Hence, R Í U (2) From (1) and (2), we have U = R Theorem 2.7: A field has no proper ideals. Proof: Let F be a field and U be an ideal of R. Then we will prove that either U = {0} or U = F. From the definition of an ideal, we have U Í F. (1) Let U ¹ {0}, a Î U and a ¹ 0 a Î U Þ a Î F (since U Ì F) - Þ a 1 Î F (since F is a field) - - Now a Î U, a 1 Î F Þ aa 1 = 1 Î U Let x Î F: then x = x × 1 Î F x Î F, 1 Î U Þ x × 1 Î U (since U is an ideal)

Chapter-02 2nd Proof Anvi T12.p6547 10/5/10, 11:53 AM 48 Linear Algebra Þ x Î U Thus, F Í U (2) From (1) and (2), we have U = F Therefore, {0} and F are the only ideals of F. Theorem 2.8: A non-zero commutative ring with unity is a field if it has no proper ideals. Proof: Let R be a commutative ring with unity such that R has no proper ideals. In order to prove that every non-zero element in R has a multiplicative inverse, let a ¹ 0 Î R, then the set Ra = {ra : r Î R} is an ideal of R. R is with unity; therefore, 1 Î R, such 1 × a = a Î Ra i.e., a ¹ 0 Î Ra Þ Ra is not a zero ideal R has no proper ideals and Ra ¹ {0}, therefore, it follows that Ra = R Now 1 Î R Þ 1 Î Ra Þ there exists an element b Î R such that 1 = ba - Þ a 1 = b Thus, every non-zero element of R has a multiplicative inverse. Accordingly R is a field. Theorem 2.9: The intersection of two ideals of a ring R is an ideal of R. Î Î Solution: Let U1 and U2 be two ideals of a ring R. Then 0 U1, 0 U2 where 0 is zero element of the ring R. Î Ç We have 0 U1 U2 Î Ç Þ Ç ¹ f and, 0 U1 U2 U1 U2 Î Ç Î Let a, b U1 U2, and r R Î Ç Þ Î Î a, b U1 U2 a, b U1 and a, b U2 Î Î Þ - Î Î Now a, b U1, r R a b U1, ar, ra U1 (1)

(since U1 is an ideal of R) Î Î Þ - Î Î and, a, b U2, r R a b U2, ar, ra U2 (2)

(since U2 is an ideal of R)

Chapter-02 2nd Proof Anvi T12.p6548 10/5/10, 11:53 AM 50 Linear Algebra Î Î Þ Î È a U1, b U2 a, b U1 U2 Þ Î Î a, b U1 or a, b U2 Þ - Î - Î a b U1 or a b U2 - Î - - Î Q If a b U1, then a (a b)= b U1 ( U1 is an ideal of R) a contradiction - Î - Î Q and ifa b U2, then b + (a b)= a U2 ( U2 is an ideal of R) a contradiction Ë Ë Our assumption that U1 U2 and U2 U1 leads to a contradiction. Í Í Hence, U1 U2 or U2 U1 2.3.5 Sum of Ideals

Definition 2.6: Let U1 and U2 be two ideals of a ring R, then the set Î Î U1 + U2 = {a + b: a U1, b U2} is called the sum of ideals U1 and U2.

Theorem 2.12: If U1 and U2 are any two ideals of a ring R, then U1 + U2 is an ideal of R containing both A and B. Î ¹ f Proof: Clearly 0 = 0 + 0 U1 + U2; therefore, U1 + U2 . a b Î Î Consider = a1 + b1, = a2 + b2; a1, a2 U1, and b1, b2 U2 a - b - - Î Then, we have = (a1 a2) + (b1 b2) U1 + U2 - Î - Î Î Î (since a1 a1 U1; b1 b2 U2 for all a1, a2 U1, b1, b2 U2) further, for any r Î R, a Î r = a1r + b1r U1 + U2 a Î r = ra1 + rb1 U1 + U2 Î Î since a1r, ra1 N1, and b1r, rb1 U2 Hence, U1 + U2 is an ideal of R. Î Þ Î Now a U1 a = a + 0 U1 + U2 Þ Í Q Î U1 U1 + U2 ( 0 U2) Î Þ Î Q Î b U2 b = 0 + b U1 + U2 ( 0 U1) Þ Í U2 U1 + U2 This proves the theorem. Definition 2.7: Let S be any subset of a ring R and U be an ideal of R. Then U is said be generated by S if (i) S Í U; and (ii) for any ideal V of R S Í V Þ U Í V

Chapter-02 2nd Proof Anvi T12.p6550 10/5/10, 11:53 AM Subrings and Ideals 51 If the ideal U is generated by a subset S of a ring R, then we denote U by the symbol < S >. From the definition, < S > is the intersection of all those ideals of R which contain S.

Theorem 2.13: If U1 and U2 are any two ideals of a ring R, then U1 + U2 = < U1 È U2 >.

Proof: If U1, U2 are ideals of R, then by the previous theorem U1 + U2 is also an Í Í ideal of R, such that U1 U1 + U2 and U2 U1 + U2. Í Í Þ È Í We have U1 U1 + U2, U2 U1 + U2 U1 U2 U1 + U2 Let V be any ideal of R such that È Í U1 U2 V Î Î Î If x U1 + U2, then x = a + b, a U1, b U2 Î È Î È now a U1 U2, b U1 U2 Þ a, b Î V Þ a + b Î V Þ x Î V Í Hence, U1 + U2 V, consequently, by definition È U1 + U2 = < U1 U2 >. Example: If R is a ring with an R has no right ideals except R and {0}, show that R is a divisor ring. Solution: Let x ¹ 0 Î R Consider x R = {xr: r Î R} x Î R Þ x = x × 1 Î xR, so xR ¹ f Also xy - xz = x (y - z) Î xR " xy, xz Î xR and for any S Î R, (xr) s = x (rs) Î x R. Thus, xR is a right ideal of R. Since x ¹ 0 Î xR, xR = R. Since R is a ring with unity, there exists y Î R such that xy = 1. Thus, R - {0} is a semigroup with unity and every non-zero element of R - {0} is right invertible. Hence, R - {0} is a group under multiplication. Consequently, R is a divisor ring.

2.4 SIMPLE RING

Definition 2.8 A ring R is said to be simple, if (i) there exist a, b Î R such that ab ¹ 0, and, (ii) R has no proper ideals. Theorem 2.14: A divisor ring is a simple ring.

Chapter-02 2nd Proof Anvi T12.p6551 10/5/10, 11:53 AM Subrings and Ideals 53 Þ Î r1 a V (Since U is an ideal) Þ x Î V Thus, U Ì V Hence, U is an ideal of R generated by a, i.e., U is a principal ideal. Example: (Z, +, ×) is a commutative ring with unity. E = < 2 > = {2n : n Î z} is an ideal of Z generated by 2 Thus E is a principal ideal of Z.

2.6 PRINCIPAL IDEAL RING

Definition 2.10: A ring for which every ideal is a principal ideal is called a principal ideal ring.

Example: The ring (Z5, +5, x5) is a principal ideal ring. Theorem 2.16: Every field is a principal ideal ring. Proof: Let F be a field. We know, that a field has no proper zero divisor, i.e., the null ideal and the unit ideal are the only ideals of F. The null ideal U = {0} = < 0 > is generated by 0 and the unit ideal F = < 1 > is generated by, therefore U = < 0 > and F are the principal ideals of F. i.e., every ideal of F, is a principal ideal. Hence, F is a principal ideal ring. ´ Example: Find the principal ideals of the ring (Z6, +6, 6)

Solution: We have Z6 = {0, 1, 2, 3, 4, 5} ´ Clearly (Z6, +6, 6) is a commutative ring with unity and it is not a field. Q ( Z6 is a ring with zero divisor)

(0) = {0} the null ideal is a principal ideal of Z6

(1) = z6, the unit ideal is a principal ideal of the ring (2) = {0, 2, 4} is a principal ideal of the ring (3) = {0, 3} is a principal ideal of the ring (4) = (2) is a principal ideal (5) = (1) is a principal ideal ´ Hence, the principal ideals of (Z6, +6, 6) are (0), (1), {0, 3}, {0, 2, 4} Theorem 2.17: The ring of integers Z is a principal ideal ring. Proof: The ring of integers (Z, + , ×) is a commutative ring with unity and with- out zero divisors; therefore, (Z, +, ×) is an integral domain. Let U be an ideal of Z. If U = {0} = < 0 >, then U is a principal ideal Let us suppose that U ¹ (0).

Chapter-02 2nd Proof Anvi T12.p6553 10/5/10, 11:53 AM 54 Linear Algebra U contains at least one non-zero integer say a. Since U is a subgroup Z under addition; a Î U Þ - a Î U, where one of the integers, a, - a is positive. Hence, we conclude that U contains at least one positive integer. Let U+ denote the set of all positive integers of U. From the well ordering principle, U+ must have at least element. Let b denote least element in U+. We now show that U = (b); i.e., U is a principal ideal generated by b. Let x Î U, then x, b are integers where b ¹ 0. There exist integers q, r such that x = bq + r, 0 £ r < b U is an ideal, therefore, b Î U, q Î Z Þ bq Î U also x Î U, bq Î U Þ x - bq = r Î U(Since U is a subgroup of (Z, +)) but 0 £ r < b and b is the least positive integer such that b Î U implies that r < b and r Î U, which is a contradiction. Therefore, r must be zero. Hence, x = bq Therefore, U = {bq : q Î z} = (b) i.e., U is a principal ideal of Z, generated by b. Thus, every ideal of Z is a principal ideal. Hence, the ring of integers is a principal ideal ring. Example: If R is a commutative ring and a Î R, then the principal ideal (a) is equal to the set {ar + na : r Î R, n Î Z} Solution: Let U = {ar + na : r Î R, n Î Z)} We shall prove that U is the ideal generated by a. Clearly a = a0 + 1a Î U so that U ¹ f Let ar + na, as + ma Î U, where r, s Î R, n, m Î Z Then (ar + na) - (as + ma)= a (r - s) + (n - m) a Î U Further, if s Î R, then (ar + na)s = a (rs) + a (ns) = a (rs + ns) + 0a Î U which shows that U is a right ideal of R. Since R is commutative, U is also a left ideal of R. Hence, U is an ideal of R such that (a) Ì U. Let V be another ideal of R, containing (a). Since a Î V and V is an ideal of R, ar Î V, n a Î V for all r Î R and n Î Z. Therefore, ar + na Î V for all r Î R and for all n Î Z. Hence, U Í V consequently, U = (a)

Chapter-02 2nd Proof Anvi T12.p6554 10/5/10, 11:53 AM 56 Linear Algebra = [r + U] + [(s + t) + U] = (r + U) + [(s + U) + (t + U) Thus, addition is associative in R/U Existence of Identity: U = 0 + U Î R/U such that (r + U) + (0 + U)= (r + 0) + U = r + U and, (0 + U) + (r + U)= (0 + r) + U = r + U for all r + U Î R/U Hence, U = 0 + U is the identity with respect to addition. Existence of inverse: r + U Î R/U Þ - r + U Î R/U such that (r + U) + (- r + U)= (r + (- r)) + U = 0 + U = U and, (- r + U) + (r + r)= (- r + r) + U = 0 + U = U Thus, each element is invertible under addition. Commutative Property: r + U, s + U Î R/U Þ (r + U) + (s + U)= (r + s) + U = (s + r) + U = (s + U) + (r + U) (Since (R, +) is abelian Þ r + s = s + r " r, s Î R) Hence, (R/U, +) is an abelian group. I. R/U is closed with respect to multiplication (by def.) Association axiom: r + U, s + U, t + U Î R/U Þ [(r + U)] (s + U) (t + U) = (rs + U) (t + U) = (rs)t + U = r (st) + U (Q r, s, t Î R Þ (rs) t = r (st) = (r + U) (st + U) = (r + U) [(s + U) (t + U)] Thus, multiplication is associative in R/U. II. Distributive Laws: We have (r + U) [(s + U) + (t + U)] = (r + U) [(s + t) + U] = (r + (s + t)] + U = (rs + rt) + U = (rs + U) + (rt + U) = (r + U) (s + U) + (r + U) (t + U)

Chapter-02 2nd Proof Anvi T12.p6556 10/5/10, 11:53 AM Subrings and Ideals 57 for all r + U, s + U, t + U Î R/U Similarly, we can prove that [(s + U) + (t + U)] (r + U) = (s + U) (r + U) + (t + U) (r + U) Hence, (R/U, +, ×) is a ring. Definition 2.11: Let R be a ring and U be any ideal of R, then the system (R/U, +, ×) where R/U = {r + U: r Î R} and ‘+’ ‘×’ are the binary operations on R/ U defined by (r + U) + (s + U)= (r + s) + U (r + U) + (s + U)= rs + U for all r, s Î R (i.e., for all r + U, S + U Î R/U) is a ring called the quotient ring of R with respect to the ideal U. Example: Let U = {6n: n Î Z}, U is an ideal of Z and Z/U = {U, 1 + U, 2 + U, 3 + U, 4 + U, 5 + U} is the quotient ring under the operations ‘*’ and ‘,’. Addition modulo 6 and mul- tiplication modulo 6. Remarks 1: If R is commutative then R/U is also commutative, since (r + U) (s + U)= rs + U = sr + U = (s + U) (r + U) (Q rs º sr " r, s Î R) 2. If R has unity element, then R/U also has unity element: Let 1 be the unity element in R, then 1 + U Î R/U such that (1 + U) × (r + U)= 1 × r + U = r + U for all r + U Î R/U (Q 1 × r = r for all r Î R) Therefore, 1 + U is the unity element in R/U.

2.8 PRIME IDEAL

Definition 2.12: Let R be a commutative ring. An ideal P of R is called a prime ideal if ab Î P Þ a Î P or b Î P for all a, b Î R

Example 1: The ideal (3) = {3n : n Î z} is a prime ideal in z, since 3 |ab Þ 3|a or 3|b Þ a Î (3) or b Î (3) In general, P = {pr; r Î Z, p is a prime} is a prime ideal of Z.

Chapter-02 2nd Proof Anvi T12.p6557 10/5/10, 11:53 AM Subrings and Ideals 59 = {... - 21, - 14, - 7, 0, 7, 14, 21, ...} is maximal ideal in z. Alternative definition: A proper ideal M of a ring R is called a maximal ideal if there does not exist an ideal U of R such that M Í U Í R. Theorem 2.20: An ideal ring Z of integers is a maximal ideal if and only if it is generated by some prime number. Proof: Let M be an ideal of Z generated by a prime number p, and let M = {pn: n Î z} = < p > Let U be any ideal of Z, such that M Ì U Ì Z Every ideal of Z is a principal ideal Thus, U = < q >, q is an integer Now M Ì U Ì Z Þ < p > Ì < q > Ì Z Þ p Î < q > Þ p = qm for some m Î Z but p is prime Þ q = 1 or m = 1 m = 1 Þ p = q Þ < p > = < q > Þ M = U q = 1 Þ < q > = z Þ U = Z Hence, M is maximal ideal. Conversely, let M be a maximal ideal in Z and let M = (p). Let us assume that p is not a prime, then p must be a composite number. There exist integers a and b such that p = ab. Let a, b be prime numbers, then U = < a > and U É M Thus M Ì U Ì Z but M is a maximal ideal Hence, M = U or U = Z Case (i) When U = z we have U = < a > = < 1 > Þ a = 1 Therefore, p = ab Þ p = b Þ p is a prime number.

Chapter-02 2nd Proof Anvi T12.p6559 10/5/10, 11:53 AM 60 Linear Algebra Case (ii) when, U = M We have U = < a > = M Þ a Î M Þ a Î rp, r Î Z Therefore, p = ab = (rp) b = p(rb) Þ 1= rb Þ r = 1, b = 1 Thus, p = a × 1 = a Þ p is prime Hence, M is generated by the prime p. Theorem 2.21: An ideal M ¹ R of a commutative ring R with unity is maximal if and only if R/M is a field. Proof: Let R be a commutative ring with unity and let M be a maximal ideal of the ring R. The R/M is also a commutative with unity. 1 + M is the unity of the ring R/M. R/M is a field, if we show that every non-zero element of R/M has a multipli- cative inverse. M is a maximal ideal of R, therefore a Î R, a Ï M Þ < a > + M = R(1) there exist elements b Î R, x Î M such that x + ab = 1 or ab - 1= x Î M If ab - 1 Î M, then we have ab + M = 1 + M Þ (a + M) (b + M) = 1 + M i.e., to each non-zero element a + M Î R/M, there exists b + M Î R/M such that (a + M) (b + M) = 1 + M Thus, a + M Î R/M is invertible. Hence, R/M is a field. Conversely, let R/M be a field and U be an ideal of U ¹ M and M Ì U We now show that U = R Since U É M, U ¹ M there exists an element a Î U such that a Ï M a Ï M Þ a + M is a non-zero element of R/M. R/M is a field and a + M is a non-zero element in R/M. Hence, a + M is invertible

Chapter-02 2nd Proof Anvi T12.p6560 10/5/10, 11:53 AM 62 Linear Algebra Solution: Let a + U = b + U 0 Î U Þ a = a + 0 Î a + U = b + U now at b + U Þ a = b + x for some x Î U Þ a - b = x Î U Conversely, let a - b Î U; and a - b = c, then a - b = c Î U Þ a = b + c we have x Î a + U Þ x = a + d for some d Î U Hence, x = (b + c) + d = b + (c + d) Î b + 0 (Q c + d Î U) Thus, a + U Ì b + U Similarly, b + U Ì a + U Therefore, a + U = b + U

Chapter-02 2nd Proof Anvi T12.p6562 10/5/10, 11:53 AM