Section 18 -- Rings and Fields

Section 18 – Rings and ﬁelds

Instructor: Yifan Yang

Spring 2007 Motivation

• Many sets in mathematics have two binary operations (and thus two algebraic structures)

• For example, the sets Z, Q, R, Mn(R) (the set of n × n matrices) all have addition and multiplication. • Note that the multiplication in Z, Q, and R are commutative, while that of Mn(R) is not. • Also, every non-zero element of Q and R has an multiplicative inverse, but this is not the case for Z and Mn(R). • We wish to give an axiomatic study on these subjects, and determine how various properties (such as commutativity of multiplication, the existence of multiplicative inverse and so on) affect the overall algebraic structures of the sets. Rings and ﬁelds

Deﬁnition A ring hR, +, ·i is a set R together with two binary operations + and ·, called addition and multiplication, such that the following axioms are satisﬁed: 1. hR, +i is an abelian group. 2. Multiplication is associative, i.e., a(bc) = (ab)c for all a, b, c ∈ R. 3. For all a, b, c ∈ R, the left distributive law a · (b + c) = (a · b) + (a · c) and the right distributive law (a + b) · c = (a · c) + (b · c) hold. Examples

• The set Z, Q, R, and C are all rings with the usual addition and multiplication.

• The set Zn of all residue classes modulo n is a ring with ¯ ¯ ¯ a¯ +n b = a + b and a¯ ·n b = ab. • The set Mn(R) of all n × n matrices is a ring. • Let R1 and R2 be rings. Deﬁne + and · on R1 × R2 by

(a1, b1) + (a2, b2) = (a1 + a2, b1 + b2)

and (a1, b1) · (a2, b2) = (a1 · a2, b1 · b2).

Then R1 × R2 is a ring, called the direct product of R1 and R2. Examples

• Let F be the set of all continuous functions f : R → R. Deﬁne f + g and f · g to be

f + g : x 7→ f (x) + g(x), f · g : x 7→ f (x)g(x)

Then hF, +, ·i is ring. • Let F be the set of all linear transformations from Rn to Rn. Let addition and multiplication be deﬁned by

f + g : v 7→ f (v) + g(v), f ◦ g : v 7→ f (g(v))

Then hF, +, ◦i is a ring. (Actually, this is just a different way to say that Mn(R) is a ring.) Notation

• For simplicity, we write ab in place of a · b. • The additive identity of a ring R is denoted by 0. In case some confusion may arise, we also write 0R. • For an element a of R, we let −a denote the additive inverse of a. • For a positive integer n and an element a of R, the notation n · a refers to the sum a + ··· + a having n summands.

• By convention we set 0 · a = 0R. Here the 0 on the left-hand side is the integer 0, while 0R on the right is the additive identity element of R. Basic properties

Theorem (18.8) Let R be a ring. For any a, b ∈ R, we have

1. 0Ra = a0R = 0R. 2. a(−b) = (−a)b = −ab. 3. (−a)(−b) = ab.

Proof of (1). Since 0R + 0R = 0R, we have (0R + 0R)a = 0Ra. Then by the distributive law, we have

0Ra + 0Ra = 0Ra.

Then using the cancellation law for groups, we obtain

0Ra = 0R.

The proof of a0R = 0R is similar. Proof of Theorem 18.8, continued

Proof of a(−b) = −ab = (−a)b. We have, by deﬁnition of −b,

b + (−b) = 0R.

By (1), ab + a(−b) = a0R = 0R. This means that a(−b) is the additive inverse of ab. That is, a(−b) = −ab. The proof of (−a)b = −ab is similar. Proof of Theorem 18.8, continued

Proof of (−a)(−b) = ab. By (2) (−a)(−b) = −(a(−b)). By (2) again, −(a(−b)) = −(−ab). This means that (−a)(−b) is the additive inverse of −ab. But we know that the additive inverse of −ab is ab. Thus, by the uniqueness of additive inverse, we conclude(−a)(−b) = ab. In-class exercises

Determine whether the following algebraic structures are rings. 1. The set Z[x] of all polynomials over Z. 2. The set GLn(R) of all n × n invertible matrices under the usual matrix addition and multiplication. a b 3. The set : a, b ∈ under the usual matrix −b a R addition and multiplication. 1 4. The set 2 Z = {n/2 : n ∈ Z} under the usual addition and multiplication. 5. The set n a a o [1/2] = a + 1 + ··· + n : a ∈ Z 0 2 2n i Z under the usual addition and multiplication. Homomorphisms

Deﬁnition For rings R and R0, a function φ : R → R0 is a (ring) homomorphism if 1. φ(a + b) = φ(a) + φ(b), 2. φ(ab) = φ(a)φ(b), for all a, b ∈ R. The kernel of φ is the set

Ker(φ) = {a ∈ R : φ(a) = 0}. Examples

0 • Let n be a positive integer. Let R = Z and R = Zn. Then the function φ : Z → Zn deﬁned by

φ(a) = a mod n

(or φ(a) = a + nZ in our notation from the last semester) is a ring homomorphism. • The function ψ : Z → 2Z deﬁned by

ψ(a) = 2a

is not a ring homomorphism since ψ(ab) = 2ab, but ψ(a)ψ(b) = 4ab. Note that if we consider Z and 2Z as additive groups, then ψ is a group homomorphism. Example

Let R = R[x] be the set of all polynomials over R. Given a ∈ R, deﬁne φa : R[x] → R by

φa(f (x)) = f (a).

Then φa is an evaluation homomorphism. Isomorphisms

Deﬁnition Let R and R0 be two rings. A function φ : R → R0 is an isomorphism if 1. φ is a ring homomorphism, 2. φ is one-to-one, or equivalently, Ker(φ) = {0}, 3. φ is onto. We then say R and R0 are isomorphic. Example

a b The map φ : → : a, b ∈ deﬁned by C −b a R

a b φ(a + bi) = −b a

is a ring isomorphism. The map

a −b ψ(a + bi) = b a

is another isomorphism. Multiplicative deﬁnitions

Deﬁnition • A ring in which the multiplication is commutative is a commutative ring. • If an element a of R satisﬁes

ra = ar = r

for all r ∈ R, then a is the multiplicative identity or the unity, and is denoted by 1 or 1R. • A ring with a multiplicative identity is a ring with unity. • A multiplicative inverse of an element a in a ring with unity is an element a−1 of R such that a−1a = aa−1 = 1. Multiplicative deﬁnitions

Definition Let R be a ring with unity 1 6= 0. • An element u of R is a unit if it has a multiplicative inverse. • If every nonzero element of R is a unit, then R is a division ring (or skew field). • A commutative division ring is a field. • A noncommutative division ring is a strictly skew field. Remarks

• If a ring R has a multiplicative identity element, it is unique. (See the proof of Theorem 3.13.) • If an element a of a ring with unity has a multiplicative inverse, the inverse is unique. • The assumption 1 6= 0 in the deﬁnition of a ﬁeld is to avoid the trivial case where the ring consists of just one single element 0. (In this case 0 is both the additive identity and the multiplicative identity.) • Conversely, if 1 = 0, then for all a ∈ R, a = 1a = 0a = 0, and R = {0} is the trivial ring. Examples

• The sets Z, Q, R, and C are all commutative rings with unity under usual addition and multiplication. • The rings Q, R, and C are ﬁelds, while Z is not since only ±1 are units in Z. • The set Mn(R) of all n × n matrices is a noncommutative ring. The set of units in Mn(R) is GLn(R). The set Mn(R) is not a division ring since there exist nonzero matrices that are not invertible.

• The set Z2, Z3, and Z5 are ﬁelds. (Take Z5 for example, the multiplicative inverses of 1¯, 2¯, 3¯, 4¯ are 1¯, 3¯, 2¯, 4,¯ respectively.) • If gcd(a, n) > 1, then a¯ never has a multiplicative inverse in ¯ Zn, since a¯b will be a multiple of gcd(a, n) for any b. This shows that if n is composite, then Zn is never a ﬁeld. In-class exercises

1. Find the units in Z12. 2. Find the units in Z15. 3. Let F be the ring of all continuous functions f : R → R with addition and multiplication given by

f + g : x 7→ f (x) + g(x), f · g : x 7→ f (x)g(x).

What are the units in F? Subrings and subﬁelds

Definition A subset S of a ring R is a subring of R if S is a ring under the induced addition and multiplication from R. We denote this relation by S < R.A subfield is similarly defined. Homework

Problems 8, 10, 12, 18, 19, 24, 25, 37, 40, 48, 50 of Section 18.