Chapter 3

Suppose that \(S\) is a set and \(R\) is a ring. Let \(R^{S}\) denote the set of all mappings from \(S\) to \(R\). Show that \(R^{S}\) is a ring, under the operations defined by $$ (f+g)(s)=f(s)+g(s), \quad(f g)(s)=f(s) g(s) $$ Show that if \(S\) has more than one element then there exist non-zero elements \(f\) and \(g\) in \(R^{S}\) for which \(f g=0\).

Suppose that \(R\) is an integral domain, with field of fractions \(F\). Show that the field of fractions of \(R\left[x_{1}, \ldots, x_{n}\right]\) can be identified naturally with \(F\left(x_{1}, \ldots, x_{n}\right)\).

Show that an integral domain with a finite number of elements is always a field.

Suppose that \(R\) is an integral domain with characteristic \(k\). Show that, when \(R\) is considered as an additive group, every non-zero element has order \(k\) (if \(k>0\) ) or infinite order (if \(k=0\) ).

Suppose that \(R\) is an integral domain of characteristic \(k>0\). Show how \(R\) can be considered as a vector space over \(\mathbb{Z}_{k}\).

Construct for each positive integer \(n\) an ideal in \(\mathbb{Z}[x]\) which is generated by \(n\) elements and is not generated by fewer than \(n\) elements.

Suppose that \(K\) is a field. If \(f=a_{0}+a_{1} x+\cdots+a_{n} x^{n} \in K[x]\) and \(k \in K\), let \((\phi(f))(k)=a_{0}+a_{1} k+\cdots+a_{n} k^{n} .\) Show that \(\phi\) is a ring homomorphism from \(K[x]\) to \(K^{K}\). Show that if \(K\) is finite then \(\phi\) is an epimorphism, but not a monomorphism. What happens if \(K\) is infinite?

Suppose that \(R\) is an infinite ring such that \(R / I\) is finite for each non- trivial ideal \(I .\) Show that \(R\) is an integral domain.

What are the units in \(R[x]\), where \(R\) is an integral domain? What are the units in \(\mathbb{Z}_{4}[x]\) ?

Show that an element \(a\) of an integral domain \(R\) is prime if and only if \(R /(a)\) is an integral domain.

Suppose that \(f=k_{0}+k_{1} x+\cdots+k_{n} x^{n}\) is a non-zero element of \(K[x]\) (where \(K\) is a field). An element \(\alpha\) of \(K\) is a root of \(f\) if \(f(\alpha)=\) \(k_{0}+k_{1} \alpha+\cdots+k_{n} \alpha^{n}=0 .\) Show that \(\alpha\) is a root of \(f\) if and only if \(f \in(x-\alpha)\), and show that \(f\) has at most \(n\) distinct roots.

What are the ideals in \(\mathbb{Z}_{6} ?\) Is it a principal ideal domain?

Suppose that \(p\) is a prime number. Let \(R\) be the set of rationals which can be written in the form \(r / s\), where \(p\) does not divide \(s\). Show that \(S\) is a subring of \(\mathbb{Q}\). What are the units in \(R\) ? Show that \(R\) is a principal ideal domain.

Suppose that \(R\) is a principal ideal domain which is not a field. Show that \(R[x]\) is not a principal ideal domain.

Suppose that \(R\) is an integral domain. Show that the following are equivalent: (i) every finite non-empty set of non-zero elements of \(R\) has a highest common factor; (ii) every finite non-empty set of non-zero elements of \(R\) has a least common multiple.

Suppose that \(R\) is an integral domain with the property that every non-empty set \(B\) of non-zero elements has a highest common factor of the form \(\gamma_{1} b_{1}+\cdots+\gamma_{n} b_{n}\), with \(b_{1}, \ldots, b_{n}\) in \(B\) and \(\gamma_{1}, \ldots, \gamma_{n}\) in \(R\). Show that \(R\) is a principal ideal domain.

Show that an element of a ring \(R\) is invertible if and only if it is contained in no maximal proper ideal in \(R\).

Suppose that \(J\) is a proper prime ideal in an integral domain \(R\). Show that \(J[x]\) is prime in \(R[x]\). Show that \(J[x]\) is not a maximal proper ideal in \(R[x]\).