Higher Algebra I — Fall 2005

Gerald Hoehn

Problem sheet 8 October 13, 2005

Problem 1: A nilpotent element of a ring is an element a such that a^{n} = 0 for some nonnegative integer n.

(a) Show that the set of nilpotent elements of a commutative ring form an ideal — called the nilradical.

(b) Let R be a commutative ring with 1 ≠ 0. Prove that if a ∈ R is nilpotent, 1 − ab is a unit for all b ∈ R.

Problem 2: Let I, J and K be ideals of a ring R. Prove:

(a) I + J is the smallest ideal of R containing both I and J.

(b) IJ ⊂ I ∩ J.

(c) I(J + K) = IJ + IK and (J + K)I = JI + KI.

(d) If J ⊂ I then I ∩ (J + K) = J + (I ∩ K).

Problem 3: For an ideal I of a ring R denote with M_{n}(I) the set of n × nmatrices with entries in I. Prove that I ↦ M_{n}(I) defines a bijective map of the set of ideals of R onto the set of ideals of the ring M_{n}(R).

Problem 4: Show that every ring R can be embedded into a ring S with 1. (Hint: Take S = Z × R and define for m, n ∈ Z and a, b ∈ R the addition by (m, a) + (n, b) = (m + n, a + b) and the multiplication by (m, a)(n, b) = (mn, mb + na + ab), where for a nonegative integer n one writes na =

\underbrace{a + ⋯ + a}_{n−times}.)