[...]... algebras and quasi-Frobenius rings The class of quasi-Frobenius rings was introduced by T.Nakayama in 1939 as a generalization of Frobenius algebras It is one of the most interesting and intensively studied classes of Artinian rings Frobenius algebras are determined by the requirement that right and left regular modules are equivalent And quasi-Frobenius algebras are defined as algebras for which regular modules. .. Quasi-Frobenius rings are also of interest because of the presence of a duality between the categories of left and right finitely generated modules over them The main properties of duality in Noetherian rings are considered in section 4.10 Semiperfect rings with duality for simple modules are studied in section 4.11 The equivalent definitions of quasi-Frobenius rings in terms of duality and semiinjective rings. .. serial rings in terms of their quivers We also describe the structure of particular classes of right serial rings, suchas quasi-Frobenius rings, right hereditary rings, and semiprime xii PREFACE rings In section 5.6 we introduce right serial quivers and trees and give their description The last section of this chapter is devoted to the Cartan determinant conjecture for right Artinian right serial rings. .. called ALGEBRAS, RINGS AND MODULES 4 its Cayley table Such a table is a square array with the rows and columns labelled by the elements of the group In this table at the intersection of the i-th row and the j-th column we write the product of the elements, which are in the i-th row and the j-th column respectively It is obvious, that this table is symmetric with respect to the main diagonal if and only... ∈ X 2 Let G be any group and let X = G Define a map from G × X to X by g · x = gx, for each g ∈ G and x ∈ X, where gx on the right hand side is the product of g and x in the group G This gives a group action of G on itself This action is called the left regular action of G on itself 3 The additive group Z acts on itself by z · a = z + a, for all z, a ∈ Z ALGEBRAS, RINGS AND MODULES 16 The set StG (x)... τ 2 = 1 and στ = τ σ −1 , where σ is the cyclic permutation (1, 2, 3) and τ is the reflection (1, 3) Labeling the vertices of the triangle as 1, 2 and 3 permits us to identify the symmetries with permutations of the vertices, and we see that there are three rotation symmetries (through angles of 0, 2π/3 and 4π/3) corresponding to the identity permutation, the cycles (1, 2, 3) and (1, 3, 2), and three... m, where p is a prime and (p, m) = 1 Let P and Q be Sylow p-subgroups of G, that is, |P | = |Q| = pn Consider R = Q ∩ NG (P ) Obviously, Q ∩ P ⊆ R In addition, since R ⊆ NG (P ), RP ALGEBRAS, RINGS AND MODULES 20 |R| · |P | , by the first isomorphism theorem (see theorem |R ∩ P | 1.3.3, vol.I) Since P is a subgroup of RP , pn divides its order |RP | But R is a subgroup of Q, and |P | = pn , so |R|... Galois theory and is connected with the problem of solvability of algebraic equations in radicals Let f be a polynomial in x over a field k and K be the (minimal) splitting field of f The group Gal(K/k) is called the Galois group of f The main result of Galois theory says that the equation f (x) = 0 is solvable in radicals if and only if the group Gal(K/k) is solvable ALGEBRAS, RINGS AND MODULES 22 We... For h ∈ H, we have that g −1 hg ∈ H if and only if h−1 g −1 hg ∈ H, so that H G if and only if [h, g] ∈ H for all h ∈ H and all g ∈ G Thus, H G if and only if [H, G] ⊆ H (3) Let xG and yG be arbitrary elements of G/G By the definition of the group operation in G/G and since [x, y] ∈ G we have (xG )(yG ) = (xy)G = (yx[x, y])G = (yx)G = (yG )(xG ) (4) Suppose H G and G/H is Abelian Then for all x, y ∈... to the other three elements of S3 ALGEBRAS, RINGS AND MODULES 8 Example 1.2.2 Dihedral groups For each n ∈ Z+ , n ≥ 3, let Dn be a set of all symmetries of an n-sided regular polygon There are n rotation symmetries, through angles 2kπ/n, where k ∈ {0, 1, 2, , n − 1}, and there are n reflection symmetries, in the n lines which are bisectors of the internal angles and/ or perpendicular bisectors of the . y0 w0 h0" alt="" Algebras, Rings and Modules Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science,. Amsterdam, The Netherlands Volume 586 Algebras, Rings and Modules Vo l u m e 2 by Michiel Hazewinkel CWI, Amsterdam, The Netherlands Nadiya Gubareni Technical