The present paper is a review of some of the author's results related to multiplication modules over noncommutative rings. All rings are assumed to be associative and with nonzero identity element; all modules are unital. A ring is said to be right (resp., left) invariant if all right (resp., left) ideals of it are ideals. Expressions such as an "invariant ring" mean that the corresponding right and left conditions hold. A right (resp., left) A-module M is said to be a multiplication module if for each of its submodules N, there exists an ideal B of the ring A such that N = MB (resp., N = BM). For brevity, a ring A is called a right (resp., left) m-ring if all ideals of it are multiplication right (resp., left) A-modules. The class of all m-rings contains all invariant principal ideal rings (see Remark 1), all invariant hereditary domains (see Remark 4), and all strongly regular rings (see Remark 5). In particular, all factor rings of the polynomial ring in one variable over a field, all factor rings of direct products of division rings, and all rings of algebraic integers are examples of m-rings. There are many works containing results on commutative m-rings. The study of commutative m-rings was begun in the works of Krull and Mori [11, 12, 15-20]. Later, many authors studied commutative m-rings (e.g., see [2-7, 9, 10, 13, 14, 21, 22, 37] and [38]). Some generalizations of commutative m-rings to the case of noncommutative rings are studied in [24, 25, 32-35] and [36].
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