91Ó°ÊÓ

Prove that: (a) \(x^{2}+x+1\) is irreducible over \(F\), the field of integers mod 2 . (b) \(x^{2}+1\) is irreducible over the integers mod 7 . (c) \(x^{3}-9\) is irreducible over the integers \(\bmod 31\). (d) \(x^{3}-9\) is redueible over the integers mod 11 .

Let \(R\) be a ring such that the only right ideals of \(R\) are \((0)\) and \(R\). Prove that either \(R\) is a division ring or that \(R\) is a ring with a prime number of elements in which \(a b=0\) for every \(a, b \in R\).

If \(R\) is a commutative ring and \(a \in R\) show that: (a) \(a R=\\{\operatorname{ar} \mid r \in R\\}\) is a two-sided ideal of \(R\). (b) Show by an example that this may be false if \(R\) is not commutative.

If \(U, V\) are ideals of \(R\) let \(U V\) be the set of all elements that can be written as finite sums of elements of the form \(w\) where \(u \in U\) and \(v \in V\). Prove that \(U V\) is an ideal of \(R\).

Let \(R\) be a ring with unit element. Using its elements we define \(a\) ring \(R\) by defining \(a \oplus b=a+b+1\), and \(a \cdot b=a b+a+b\) where \(a, b \in R\) and where the addition and multiplication on the right-hand side of these relations are those of \(R\). (a) Prove that \(R\) is a ring under the operations \(\oplus\) and \(\because\) (b) What acts as the 0-element of \(R\) ? (c) What acts as the 1-element of \(R ?\) (d) Prove that \(R\) is isomorphic to \(R\).