By Michiel Hazewinkel, Nadiya M. Gubareni

The concept of algebras, jewelry, and modules is likely one of the basic domain names of contemporary arithmetic. common algebra, extra in particular non-commutative algebra, is poised for significant advances within the twenty-first century (together with and in interplay with combinatorics), simply as topology, research, and likelihood skilled within the 20th century. This quantity is a continuation and an in-depth research, stressing the non-commutative nature of the 1st volumes of **Algebras, jewelry and Modules** via M. Hazewinkel, N. Gubareni, and V. V. Kirichenko. it's mostly autonomous of the opposite volumes. The appropriate structures and effects from past volumes were awarded during this quantity.

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If the set I is finite, |I | = n < ∞, and the category is additive, then π1 σ1 + · · · + πn σ n = 1 X and the product n n X i is equal to their coproduct i=1 n this case this common object is called the direct sum and denoted by X i . In i=1 X i . Point i=1 out that this is not the case in a category of rings. The direct product of a finite number of rings were considered in [146]. Now consider this construction for an infinite number of rings. 2. Let { Ai }i ∈I , be a family of rings. The direct product of the rings { Ai }i ∈I is the Cartesian product A = Ai with operations of addition and i ∈I multiplication defined componentwise: ( f + g)(i) = f (i) + g(i) ( f g)(i) = f (i)g(i) for any f , g ∈ A and for each i ∈ I.

Conversely, all rings of these forms are serial and right Noetherian. A full description of serial right hereditary rings is given by the following theorem. 6. ) Any serial right hereditary ring A is Morita equivalent to a direct product of rings isomorphic to rings of upper 7Other equivalent definitions of a discrete valuation ring is given in Chapter 3. 6. 9Recall that two rings are said to be Morita equivalent if their categories of modules are equivalent. g. ) 33 Preliminaries triangular matrices over divisions rings, rings of the form Hm (O) and rings of the form H (O, m, n), where O is a discrete valuation ring.

Any principal ideal domain is hereditary. 3. (Kaplansky’s Theorem). g. ) If a ring is a right hereditary then any submodule of a free A-module is isomorphic to a direct sum of right ideals of A. A ring A is said to be right (left) semihereditary if each right (left) finitely generated ideal of A is a projective A-module. If a ring A is both right and left semihereditary, it is called semihereditary. Preliminaries 27 The following theorem gives some of other equivalent condition for a ring to be right (left) semihereditary.