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(c)Karen E. Smith 2018 UM Math Dept licensed under a Creative Commons By-NC-SA 4.0 International License. Math 412. Examples and Types of Rings and their

DEFINITION:A domain is a commutative R which has no (non-zero) zero-divisors.1

DEFINITION:A field is a R in which 0R 6= 1R in which every non-zero element has a multiplicative inverse.

DEFINITION:A subring2 S of a ring R is a subset of R satisfying the following

(1) 1R ∈ S; (2) S is closed under multiplication (i.e. x, y ∈ S implies x × y ∈ S) (3) S is closed under addition (x, y ∈ S implies x + y ∈ S) and taking additive inverses (i.e. x ∈ S implies −x ∈ S). Equivalently, a S of a ring R as a subset which is a also a ring with the same +, ×, 0 and 1.

“DEFINITION:” Fix a commutative ring R. The ring over R is the set n R[x] = {a0 + a1x + ··· + anx | ai ∈ R, n ∈ N}, with operations + and × extended from those on the coefficients in R in the natural way.

THEOREM 4.3: Fix a ring R. Then the polynomial R[x] is a domain if and only if R is a domain.

THEOREM 4.5: For any domain R, the units in R[x] are the units in the subring R of constant . In particular, if F is a field, then the units in F[x] are the non-zero constant polynomials.

A.WARM-UP: For each inclusion S ⊂ R, state whether or not S is a subring of R. (1) Z ⊂ Q, Z ⊂ R, R ⊂ C. (2) N ⊂ Z, (3) The set of even S = {2n | n ∈ Z} ⊂ Z. f(x) (4) R[x] ⊂ R(x) := { g(x) |f(x), g(x) ∈ R[x], g 6= 0} (the set of rational functions). a 0 (5) The set of diagonal matrices D := { | a, b, ∈ } ⊂ M ( ). 0 b R 2 R (6) The set of matrices M2(Z) ⊂ M2(R). a b (7) The set of 1 matrices GL ( ) = { | ad − bc = 1, ∈ } ⊂ M ( ). 2 R c d R 2 R (8) The set of constant polynomials R ⊂ R[x]. (9) The set of polynomials with integer coefficients Z[x] ⊂ R[x]. (10) Z ⊂ Z[i] (11) The imaginary integers Zi = {ni | n ∈ Z} ⊂ Z[i].

B.FINDANEXAMPLEOF: (1) A non-commutative ring, with a commutative subring. (2) An infinite ring with a finite subring. (3) A field which has a subring which is not a field.

C.BASIC PROOFS.

(1) Prove that if S ⊂ R is a subring, then 0R ∈ S and is the additive identity for S. (2) Prove that every field is a domain. (3) Prove that a subring of a field is a domain. Is the converse true?

1“Domain” and “” are the exact same thing. The book uses “integral domain,” which is an older word. 2Caution! This definition differs from the book’s because they do not assume rings contain a multiplicative identity! 2

(4) Let S be a subset of a ring R. Prove that S is a subring if and only if the inclusion map S,→ R sending s 7→ s is a ring . For this, think carefully about the meaning of the symbols you are using in different contexts. D. PRACTICE. Answer the following questions, using Theorem 4.3 and 4.5 above where appropriate. (1) What is the zero and one in Z3[x]? Is Z3[x] commutative? Is it a domain? (2) Use a theorem to investigate if Z72[x] a domain? If not, find a pair of non-zero elements in Z72[x] whose product is zero. 2 3 3 (3) In Z8: for f = (1 + 3x) and g = (2x + 4x ), compute and simplify f + 4g and (3x) + g. Here, we abuse notation by representing congruence classes by any integer representative. (4) How many polynomials of degree less than 3 in the ring Z2[x]? (5) How many units in Z[x]? (6) Suppose that f ∈ Q[x] has degree 5. Find the degrees of the following polynomials: f −x, f 2, f + 4x51, f − 2x5, (x2 + 1)f 3. 7 16 15 (7) Find the leading coefficient of h where h = 5x + 3x + 1 ∈ Z7[x]. 2 (8) Does x + 1 have a multiplicative inverse in Z2[x]? (9) Is there any field F such that F[x] is a field? (10) In Z8[x], compute (1 + 4x)(1 − 4x). Is the hypothesis that R is a domain necessary in Theorem 4.5?

E.PROOFOF THEOREM 4.5 Let R be a domain. Consider R as the subring of constant polyno- mials in R[x]. (1) Show that any in R is a unit in R[x]. (2) Explain why, for any two polynomials f, g ∈ R[x], deg(fg) = deg f + deg g. Is this true even when R is not a domain? (3) Prove that if f ∈ R[x] is a unit, then f is a constant polynomial. (4) Prove Theorem 4.5 (5) Find a formula for the number of units in Zp[x] where p is prime.

F. HOMOMORPHISMS. Let F be an arbitrary field. Fix any a ∈ F. (1) Prove that the map F[x] → F sending f(x) 7→ f(a) is surjective homomorphism. Write the in set-builder notation. (2) Find an injective homomorphism F → M2(F). Also find an injective map F → M2(F) which is not a ring homomorphism. (3) Is the map M2(F) → F × F sending a matrix A to its first row a ring homomorphism?

G.THE DIVISION ALGORITHMFOR POLYNOMIALS. In high school, you learned to divide polynomials to find a quotient and remainder. This works for polynomials over any field. Let’s practice. (1) Let f = x3 + 4x2 + x + 1 in R[x]. Find q and r so that f = qx2 + r, where deg r < 2. 5 3 2 2 (2) In the ring F2[x], divide the polynomial x + 3x + x + 1 by x + 1. What is the quotient and remainder? (3) In the ring Q[x], divide x4 + 3x3 − x2 + 5 by x + 1. What are the quotient q and remainder r?

BONUS: State and prove the Division algorithm for F[x], where F is a field. It might be helpful to try to remember how you proved it for Z. Hint: you can use the well-ordering axiom on the degrees of polynomials.