Discussion
Consider the following statements :
(I) A p-sylow subgroup of the underlying additive group of a finite commutative ring R with unity is an ideal of R.
(II) If M is a maximal ideal of a commutative ring R and R* be the group of units of R. Such that R is a disjoint union of M and R*, then R has a maximal ideal other than M.
Which is true ?
Answer: A
Only (I) is true.
(I) The p-Sylow subgroup of the additive group of a finite ring is \(S = \{x : p^{k}x = 0\}\) for a suitable k. It is an additive subgroup, and for any r in R, \(p^{k}(rx) = r(p^{k}x) = 0\), so \(rx \in S\). Hence S is an ideal.
(II) If R is the disjoint union of M and the units R*, then every non-unit lies in M. Any proper ideal consists of non-units, so it lies inside M. Hence M is the only maximal ideal (R is local), and the claim that R has another maximal ideal is false.
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