A new characterisation of hereditary PI rings
The main object of study are Noetherian PI rings \(R\) with the following property (P): All proper Artinian homomorphic images of \(R\) are of finite representation type. One of the main results of the paper asserts that a right Noetherian PI ring \(R\) which is an order in an Artinian ring has the property (P) if and only if \(R\) is a product \(A\times S\) of an Artinian ring \(A\) of finite representation type and a semiprime hereditary ring \(S\). It is also shown that a Noetherian nonartinian PI ring \(R\) is a hereditary prime ring if and only if all proper homomorphic images of \(R\) are of finite representation type. Hence, it is concluded that a semiprime Noetherian PI ring \(R\) is hereditary if and only if \(R\) has the property (P). The author also proves the following generalisation of a well-known result of Warfield's. Let \(R\) be a Noetherian PI ring. Then every Artinian factor ring of \(R\) is serial if and only if \(R\) is a product \(A\times S\) of an Artinian serial ring \(A\) and a semiprime hereditary ring \(S\).
scientific article; zbMATH DE number 1910266
scientific article