Definitions of integral elements and quotient rings over non-commutative rings with identity

Source abstract

Let B be an associative ring with identity, A a subring of B containing the identity of B . If B is commutative then it is customary to define an element b of B to be integral over A if it satisties an equation of the form for some a 1 , a 2 , …, a n A. This definition does not generalize readily to the case when B is non-commutative. Van der Waerden ([11], p. 75) defines b ∈ B to be integral over A if all powers of b belong to a finite A -module. This definition is quite satisfactory when A satisfies the ascending chain condition for left ideals, but in the general case this type of integrity is not necessarily transitive, even when B is commutative. Krull [6] calls an element b ∈ B which satisfies the above condition almost integral over A (but he only considers the commutative case). The subset Ā of B consisting of all almost integral elements over A is called the complete integral closure of A in B . If Ā = A , A is said to be completely integrally closed in B . More recently (in [3]), Gilmer and Heinzer (see also Bourbaki, [1]) have discussed these properties in the commutative case and have shown that the complete integral closure of A in B need not be completely integrally closed in B . If B is not commutative, the set A of elements of B almost integral over A , may not even form a ring. In [5] p. 122, Jacobson uses a definition equivalent to Van der Waerden's for the non-commutative case but the definition applies only for a very restricted class of rings.