Communicated by G. E. Wall 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 satisfies an equation of the form (1) b n + a i b nl + ••• + a n = 0 for some a t , a 2 , • • -,a n e A. This definition does not generalize readily to the case when B is non-commutative. Van der Waerden ([11], p. 75) defines b e B to be integral over A if all powers of b belong to a

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... finite ^-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 be B which satisfies the above condition almost integral over A (but he only considers the commutative case). The subset A of B consisting of all almost integral elements over A is called the complete integral closure of A in B. If A = 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 2? 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. In the present paper a new definition of integrity is proposed which is transitive and is equivalent to the usual definition in the commutative case. It produces an integral closure A which is a ring and has the property that A <= A <= B. In section 2 a (presumably) new type of quotient ring is introduced which differs from that used in Jacobson [5], p. 118 or the quotient rings, discussed by Utumi [10] and Lambek [7] p. 94. Our aim is to imitate more closely the theory of commutative rings, using Jacobson's definition of an 'm-system' ([4], p. 195). We define a quotient ring A s such that A <= A s <= B where S is an m-system of A. This section is devoted almost exclusively however to the case where S is the complement in 433 use, available at https://www.cambridge.org/core/terms. https://doi.