Ore extensions over total valuation rings.
Let \(K\) be a division ring, \(K^*\) its multiplicative group, and \(V\) a total valuation ring of \(K\), i.e. a subring of \(K\), such that \(k\in V\) or \(k^{-1}\in V\), for each \(k\in K^*\). Also, let \(J(V)\) be the Jacobson radical of \(V\), \(\sigma\) an automorphism of \(V\), and \(\delta\) a (left) \(\sigma\)-derivation of \(V\). We say that \((\sigma,\delta)\) is compatible with \(V\), if \(\delta(J(V))\subseteq J(V)\). The Ore extension \(R=V[x;\sigma,\delta]\) is defined as the polynomial ring over \(V\) in an indeterminate \(x\) with the multiplication defined by the rule \(xa=\sigma(a)x+\delta(a)\), for each \(a\in V\). The quotient ring of the Ore extension \(K[x;\sigma,\delta]\) of \(K\) is denoted by \(K(x;\sigma,\delta)\). The paper under review studies the structure of \(R\) from the perspective of order theory. Note that an order in a simple Artinian ring \(Q\) means a prime Goldie ring with a quotient ring \(Q\). Let \(S\) be an order in the ring \(Q\). A left \(S\)-submodule \(I\) of \(Q\) is said to be a left \(S\)-ideal, if \(I\) contains a regular element in \(S\) and \(Iu\subseteq S\), for some \(u\in U(Q)\), where \(U(Q)\) is the group of units in \(Q\). When this holds, \(I\) is called integral, if \(I\subseteq S\). Right \(S\)-ideals are defined similarly. A right and left \(S\)-ideal is said to be an \(S\)-ideal. Let \(I\) be a left \(S\)-ideal. Then \((S:I)_r=\{q\in Q:Iq\subseteq S\}\) is a right \(S\)-ideal and \(_vI=(S:(S:I)_r)_l\) is a left \(S\)-ideal including \(I\), where \((S:J)_l=\{q\in Q:qJ\subseteq S\}\) for any right \(S\)-ideal \(J\). An order \(S\) in \(Q\) is called a left \(v\)-Bezout order, if \(_vI\) is principal, for any finitely-generated left integral \(S\)-ideal \(I\) (a right \(v\)-Bezout order is defined similarly). We say that \(S\) is a \(v\)-Bezout order, if it is a left and right \(v\)-Bezout order; \(S\) is called a Bezout order, if every finitely-generated one-sided \(S\)-ideal is principal. Clearly, Bezout orders are \(v\)-Bezout. The present paper shows that when \((\sigma,\delta)\) is compatible with \(V\), \(R\) is a \(v\)-Bezout order in \(K(x;\sigma,\delta)\), and it is not Bezout in case \(V\neq K\). It also gives necessary and sufficient conditions that \(R\) is a fully left bounded ring, i.e. for each prime ideal \(P\) of \(R\), any essential left ideal of \(R/P\) contains a nonzero ideal.
fully left bounded rings of division rings
scientific article; zbMATH DE number 5818581
scientific article