Semirings of formal power series (Q1579860)

Let \((S,+,\cdot)\) be an additively commutative semiring with absorbing zero 0 and identity \(1\not=0\). An element \(a\in S\) is called semi-invertible if there are \(s,t\in S\) such that \(1+ra=sa\) and \(1+ar=as\). The set \(U\) of all semi-invertible elements of \(S\) is a subsemigroup of \((S,\cdot)\), \(M=S\setminus U\) is the union of all proper \(k\)-ideals of \(S\) and also the union of all maximal \(k\)-ideals of \(S\). Moreover, \(S\) is called a \(k\)-division semiring if \(U=S\setminus\{0\}\) holds, which clearly implies that \(S\) has no zero-divisors. Finally, a commutative \(k\)-division semiring is called a \(k\)-semifield. Now let \(S[[x]]\) denote the semiring of all formal power series \(f=\sum_{i=0}^\infty a_ix^i,\;a_i\in S\). Then \(f\) is semi-invertible in \(S[[x]]\) iff \(a_0\) is semi-invertible in \(S\). For a \(k\)-semifield \(S\) it is shown that (i) every ideal of \(S[[x]]\) is principal, (ii) \(S[[x]]\) is a local semiring, i.e., it has a unique maximal \(k\)-ideal, and (iii) the Jacobson radical of \(S[[x]]\) is precisely \(\{f=\sum_{i=0}^\infty a_ix^i\mid a_0=0\}\).