By Michiel Hazewinkel, Nadiya M. Gubareni

The idea of algebras, earrings, and modules is likely one of the primary domain names of contemporary arithmetic. common algebra, extra particularly 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 chance skilled within the 20th century. This quantity is a continuation and an in-depth learn, stressing the non-commutative nature of the 1st volumes of **Algebras, earrings and Modules** via M. Hazewinkel, N. Gubareni, and V. V. Kirichenko. it's mostly autonomous of the opposite volumes. The proper structures and effects from prior volumes were offered during this quantity.

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A ring is right (left) serial if it is a right (left) serial module over itself. A ring which is both a right and left serial ring is said to be a serial ring. Recall that a module M is finitely presented if M is a quotient of a finitely generated free module with finitely generated kernel. The following well known theorem characterizes serial rings in terms of finitely presented modules. 1. (Drozd-Warfield Theorem, [67], [320]). g. ) For a ring A the following conditions are equivalent: 1. A is serial; 2.

Proof. 1 there exists a unique mapping ϕ : B → A such that πi ϕ = gi . So it only remains to show that ϕ is a ring morphism which follows from the fact that gi and πi are ring morphisms for each i. Indeed, for any b1 , b2 ∈ B one has πi ϕ(b1 +b2 ) = gi (b1 +b2 ) = gi (b1 )+gi (b2 ) = πi ϕ(b1 )+πi ϕ(b2 ) = πi (ϕ(b1 )+ϕ(b2 )) whence ϕ(b1 + b2 ) = ϕ(b1 ) + ϕ(b2 ). , ϕ is a ring morphism. 4. The direct product is sometimes called the complete direct product to distinguish it from the discrete direct product (or direct sum).

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. 1Recall that a category C is Abelian if its morphisms and objects can be added, there exist kernels and cokernels, and moreover, every monomorphism is a kernel of some morphism and every epimorphism is a cokernel of some morphism.