The discussion on algebraic properties of polynomial matrices
The authors show that if \(\mathbb R [t]\) is the ring of polynomials over the real field, then the ring \(M_n(\mathbb R [t])\) of \(n\times n\) matrices with entries in \(\mathbb R [t]\) is a principal and principal one-sided ideal ring. In particular they show that if \(I\) is an ideal of \(M_n(\mathbb R [t])\) then it is principal generated by a scalar matrix, and they also show that \(M_n(\mathbb R [t])\) is a left (right) Noetherian prime ring. The reviewer must point out that the previous results are known even in a more general context. We can see for instance the book of \textit{B. R. McDonald} [Linear algebra over commutative rings (1984; Zbl 0556.13003)] in particular pp.15-35 for a global overview and especially Theorem I.4 (op. cit., p.16) and the section on ``Morita correspondence'' (op.cit. Chap. III, Section B. pp.159-181). With such tools we can often lift ring-theoretic properties from \(R\) to \(M_n(R)\). For example (see op. cit. Exercise I.C.2, pp.16-17, and Exercises III.B.1-6, pp.173-181), \(R\) is a prime ring if and only if \(M_n(R)\) is a prime ring, \(R\) is left (right) Noetherian if and only if \(M_n(R)\) is left (right) Noetherian, if \(R\) has all left ideals principal, then \(M_n(R)\) has all left ideals principal, and so on. Notice that the authors also show that \(M_n({\mathbb R}[t])\) is a left Ore ring. As a final remark, we can check that all their proofs work for a polynomial ring \(K[t]\) over an arbitrary field \(K\) (and in fact instead of \(K[t]\) we can take any PID with a Euclidean algorithm).
scientific article; zbMATH DE number 1322599
scientific article