Exercise 13, § 2

Let A be a ring. (a) Give another proof of Exercise 6 (b) of Algebra, Chapter VIII, § 6, by considering the image of the polynomial $f$ in the rings $(A/p)[X]$, where $p$ runs through the set of prime ideals of A. (b) Let $N$ be a square matrix of order $r$ over A; for $N$ to be nilpotent, it is necessary and sufficient that each of the coefficients (other than the dominant coefficient) of its characteristic polynomial be so. (To see that the condition is necessary, use the same method as in (a); to see that it is sufficient, use the Cayley-Hamilton Theorem.) (c) Show that, for every ordered pair of positive integers $k, r$, there exists a least positive integer $m(k, r)$ independent of the ring A such that the relation $N^k = 0$ implies $(\mathrm{Tr}(N))^{m(k, r)} = 0$. (Consider the ring $A = \mathbf{Z}[T_{ij}]$, where the $T_{ij}$ are $r^2$ indeterminates, the matrix $X = (T_{ij})$ over A and the element $t = \sum_{i=1}^r T_{ii} = \mathrm{Tr}(X)$ of A; if $a$ is the ideal of A generated by the elements of $X^k$, show, with the aid of (b), that there exists an integer $m$ such that $t^m \in a_{k,r}$) (d) Show that $m(2, 2) = 4$.