Math 323: Homework 2 1. Section 7.2 Problem 3, Part (c) Define the set R[[x]] of formal power series in the indeterminate x with coefficients from R to be all formal infinite sums ∞ n 2 3 anx = a0 + a1x + a2x + a3x + ··· n=0 X Define addition and multiplication of power series in the same way as for power series with real or complex coefficients i.e. extend polynomial addition and multi- plication to power series as though they were “polynomials of infinite degree” ∞ ∞ ∞ n n n anx + bnx = (an + bn)x n=0 n=0 n=0 X X X ∞ ∞ ∞ n n n n anx × bnx = akbn−k x n=0 ! n=0 ! n=0 k=0 ! X X X X Part (a) asks to prove that R[[x]] is a commutative ring with 1. While you should convince yourself that this is true, do not hand in this proof. ∞ n Prove that n=0 anx is a unit in R[[x]] if and only if a0 is a unit in R. Hint: look at Part (b) first: Part (b) asks to prove that 1 − x is a unit in R[[x]] with (1 − x)−1 =1+P x + x2 + ... 2. Section 7.3 Problem 17. Let R and S be nonzero rings with identity and denote their respective identities by 1R and 1S. Let φ : R → S be a nonzero homomorphism of rings. (a) Prove that if φ(1R) =6 1S, then φ(1R) is a zero divisor in S. Deduce that if S is an integral domain, then every non-zero ring homomorphism from R to S sends the identity of R to the identity of S. −1 −1 (b) Prove that if φ(1R)=1S then φ(u) is a unit in S, and that φ(u )= φ(u) for each unit u ∈ R. 3. Section 7.3 Problem 2. Prove that the rings Z[x] and Q[x] are not isomorphic. 4. Section 7.3. Problem 6. Decide which of the following are ring homomor- phisms from M2(Z) to Z: a b (a) 7→ a (projection onto the 1, 1 entry) c d a b (b) 7→ a + d (the trace of the matrix) c d a b (c) 7→ ad − bc (the determinant of the matrix). c d 5. Section 7.3 Problem 10. Decide which of the following are ideals of Z[x]: (a) the set of all polynomials whose constant term is a multiple of 3. (b) the set of all polynomials whose coefficient of x2 is a multiple of 3. (c) the set of all polynomials whose constant term, coefficient of x, and coefficient of x2 are zero. 1 2 (d) Z[x2] (i.e. the polynomials in which only even powers of x appear). (e) the set of polynomials whose coefficients sum to zero. (f) the set of polynomials p(x) such that p′(0) = 0. 6. Section 7.3 Problem 14. Prove that the ring M4(R) contains a subring that is isomorphic to the real Hamilton Quaternions H. 7. Section 7.3 Problem 19. Prove that if I1 ⊂ I2 ⊂ · · · are ideals of R, then ∞ n=1 In is an ideal in R. S 8. Section 7.3 Problem 21.* Prove that all the two-sided ideals in the matrix ring Mn(R), where R is a ring with 1, are of the form Mn(J), where J ⊂ R is a two-sided ideal. 9. Section 7.3 Problem 24. Let φ : R → S be a ring homomorphism. (1) Prove that if J ⊂ S is an ideal, then φ−1[J] ⊂ R is an ideal. Apply this to the special case when R is a subring of S and φ is the inclusion map to deduce that if J ⊂ S is an ideal and R ⊂ S a subring, then J ∩ R ⊂ R is an ideal. (2) Prove that if φ is surjective and I ⊂ R is an ideal, then φ(I) ⊂ S is an ideal. Give an example where this fails if φ is not surjective. 10. Let Q[π] be the set of all real numbers of the form n r0 + r1π + ··· + rnπ with ri ∈ Q,n ≥ 0. It is easy to show that Q[π] is a subring or R (you don’t have to write the proof of this). Is Q[π] isomorphic to the polynomial ring Q[x]?.