On the Behrens radical of matrix rings and polynomial rings

It is shown that the Behrens radical of a polynomial ring, in either commuting or non-commuting indeterminates, has the form of “polynomials over an ideal”. Moreover, in the case of non-commuting indeterminates, for a given coefficient ring, the ideal does not depend on the cardinality of the set of indeterminates. However, in contrast to the Brown–McCoy radical, it can happen that the polynomial ring R[X]R[X] in an infinite set XX of commuting indeterminates over a ring RR is Behrens radical while the polynomial ring R〈X〉R〈X〉 in an infinite set YY of non-commuting indeterminates over RR is not Behrens radical. This is connected with the fact that the matrix rings over Behrens radical rings need not be Behrens radical. The class of Behrens radical rings, which is closed under taking matrix rings, is described.