Spring 2026: Math 831 Daily Homework

1. Let \(P\subseteq R\) be an ideal. Show that the following are equivalent:

1. (i) \(P\) is a prime ideal.
2. (ii) For all ideals \(I,J\subseteq R\), if \(IJ\subseteq P\), then \(I\subseteq P\) or \(J\subseteq P\).

Conclude that if \(P\) is a prime ideal and \(I\cap J \subseteq P\) for ideals \(I,J\subseteq R\), then \(I\subseteq P\) or \(J\subseteq P\).

2. For an ideal \(I\subseteq R\), show that:

1. (i) \(I\) is a prime ideal if and only if \(R/I\) is an integral domain.
2. (ii) \(I\) is a maximal ideal if and only if \(R/I\) is a field.

3. Let \(R[x]\) denote the polynomial ring with coefficients in \(R\). For an ideal \(I\subseteq R\), we let \(I[x]\) denote the polynomials \(f(x)\in R[x]\), all of whose coefficients are in \(I\).

1. (i) Prove that \(I[x]\) is an ideal of \(R[x]\) and equals the ideal in \(R[x]\) generated by \(I\).
2. (ii) Prove that \(R[x]/I[x]\) is isomorphic to \((R/I)[x]\).
3. (iii) Prove that if \(P\subseteq R\) is a prime ideal, then \(P[x]\) is a prime ideal in \(R[x]\).
4. (iv) Let \(M\subseteq R\) be a maximal ideal. Show that \(M[x]\) is never a maximal ideal.

1. Let \(S, T\subseteq R\) be multiplicatively closed subsets of \(R\) such that \(st \not = 0\), for all \(s\in S\) and \(t\in T\). Let \(T'\) denote the set of fractions \(\frac{t}{1}\) in \(R_S\) such that \(t\in T\). Prove that \(ST\) is a multiplicatively closed subset of \(R\), \(T'\) is a multiplicatively closed subset of \(R_S\) and \(R_{ST}\cong (R_S)_{T'}\).

2. Let \(I\subseteq R\) be an ideal. Show that there is a one-to-one correspondence between the ideals \(J\subseteq R\) containing \(I\) and the ideals \(C\subseteq R/I\). Conclude that every ideal \(C\subseteq R/I\) is of the form \(J/I\) for an ideal \(J\subseteq R\) containing \(I\). Show that this correspondence extends to prime ideals so that \(Q\subseteq R/I\) is a prime ideal if and only if \(Q = P/I\), for a prime ideal \(P\) of \(R\) containing \(I\).

3. Let \(S\subseteq R\) be a multiplicatively closed set, \(I\subseteq R\) an ideal and \(R[x]\) the polynomial ring in one variable over \(R\).

1. (i) Prove that \(R_S[x]\) is isomorphic to \(R[x]_S\).
2. (ii) Use (i), problem 2 above and problem 3 from January 20 to prove that there cannot exist a chain of prime ideals \(Q_1\subset Q_2\subset Q_3\) in \(R[x]\) contracting to same prime ideal in \(R\). Hint: Mod out \(Q_1\cap R\) to assume \(R\) is an integral domain, then localize at the complement of \(Q_1\cap R\).
3. (iii) Can you generalize the statements (i)-(ii) above to the polynomial ring in any finite set of indeterminates over \(R\)?

1. Suppose \(a\in R\) is a non-zerodivisor. Prove that \(R\) is an integral domain if and only if \(R_S\) is an integral domain, where \(S = \{1, a, a^2, \ldots \}\).

2. Let \(a \in R\) be a non-nilpotent element and set \(S := \{1, a, a^2, \ldots \}\).

1. (i) Prove that \(R_S\) is isomorphic to \(R[x]/\langle ax-1\rangle\). This is pretty tricky. Note that \(R_S = R[\frac{1}{a}]\), so there is a canonical map from the polynomial ring \(R[x]\) to \(R[\frac{1}{a}]\) such that \(ax-1\) is in the kernel. Then try to prove that if \(f(x)\) is in the kernel of the canonical map, some power of \(a\) times \(f(x)\) is a multiple of \(ax-1\) paying close attention to what that power is relative to the degree of \(f(x)\). It is helpful to think of \(f(\frac{1}{a})\) as \(\frac{g(a)}{a^n}\), for some polynomial \(g(x)\in R[x]\).
2. (ii) Use part (i) to show that if \(p \in \mathbb{Z}\) is a prime number and \(R\) denotes the set of rational numbers whose denominator is not divisible by \(p\), then \(\langle pX-1\rangle\) is a maximal ideal in \(R[X]\). Note that \(R\) is a local ring whose maximal ideal is generated by \(\frac{p}{1}\).

1. Let \(x_1, \ldots, x_n\) be indeterminates over \(R\). For \(f(x_2, \ldots, x_n) \in R[x_2,\ldots, x_n]\) and \(u\) a unit in \(R\), set \(x_1' := ux_1+f(x_2,\ldots, x_n)\). Prove that \(R[x_1, \ldots, x_n] = R[x_1',x_2, \ldots, x_n]\).

2. Let \(R\) be an integral domain, \(n\geq 2\), and \(x_1,y_1, \ldots, x_n,y_n\) indeterminates over \(R\). Prove that the ring

is an integral domain. Conclude that if \(K\) is a field and \(x,y,z,w\) are indeterminates over \(K\), then the ring \(K[x,y,z,w]/\langle xy-zw\rangle\) is an integral domain. (Hint: Use the first problem from today's assignment and the first problem from the previous assignment.)

3. Give a rigorous proof that if \(K\) is a field and \(x,y,z,w\) are indeterminates over \(K\), then the ring \(K[x,y,z,w]/\langle xy-zw\rangle\) is not a unique factorization domain.