Algebra by Dan Laksov

By Dan Laksov

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Let m be the maximal ideal pZ(p) of Z(p) . Show that the residue ring Z(p) /m is canonically isomorphic to Z/(p). 7. Let A[t] be the polynomial ring in the variable t with coefficient in A and let p be a prime ideal in A. Show that the set pA[t] of all polynomials with coefficietns in p form a prime ideal in A[t]. 8. Let K[u, v] be the polynomial ring in the two variables u and v over the field K. Is the union (u) ∪ (v) of the two ideals (u) and (v) of K[u, v] an ideal? 9. Show that if A is a ring such that 1 = 0 then A has minimal prime ideals.

2) When D(f ) ⊆ ∪α∈I D(fα ) we have that f ∈ r(a). Consequently there is a positive integer n and a finite subset J of I such that f n = β∈J fβ hβ with hβ ∈ A for all β in a finite set J . Hence f ∈ r((fβ )β∈J ) and it follows from assertion (1) n that D(f ) ⊆ ∪β∈J D(fβ ). However D(fβ ) = D(fβ β ) for all positive integers nβ . n Consequently D(f ) ⊆ ∪β∈J D(fβ β ), and using assertion (1) once more we obtain the n inclusion f ∈ r((fβ β )β∈J ) which is equivalent to the equality of assertion (2).

On the other hand we have that f1 f2 · · · fn ∈ / p, n which contradicts the assumption that ∩i=1 ai ⊆ p. 23) Exercises. 1. Let K be a field and let K[t] be the polynomial ring in the variable t over K. (1) Find all non-zero divisors in the residue class ring k[t]/(t2 ). (2) Find all the units in the residue class ring k[t]/(t2 ). 2. Let n be a positive integer. (1) Determine for which integers n the ring Z/nZ is an integral domain. (2) Determine for which integers n the ring Z/nZ is a field. 3.

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