F-algebraic extensions of rings
Let R be an algebra over a field F and A a subalgebra of R. The algebra R is F-algebraic over A if given \(r\in R\) there exists a non-constant polynomial \(p_ r(t)\in F[t]\) such that \(p_ r(r)\in A\). We prove that if R has no non-zero algebraic ideals, then (i) if A has no non-zero algebraic right ideals neither does R; (ii) if A is a domain, R is a domain; (iii) if A has no nilpotent elements, R has no nilpotent elements. We also show that if R has no non-zero algebraic right ideals neither does A and if R is prime, A is also prime. We are finally concerned with some questions about algebraic right ideals of an algebra R.
scientific article; zbMATH DE number 3843951