Serial Rings by Gennadi Puninski (auth.)

By Gennadi Puninski (auth.)

The major topic in classical ring conception is the constitution idea of earrings of a specific sort. for instance, not anyone textual content ebook in ring thought may pass over the Wedderburn-Artin theorem, which says ring R is semisimple Artinian iffR is isomorphic to a finite direct sum of complete matrix jewelry over skew fields. this is often an instance of a finiteness which, a minimum of traditionally, has ruled in ring conception. Ifwe wish to think about a demand of a lattice-theoretical variety, except being Artinian or Noetherian, the main average is uni-seriality. right here a module M is named uni-serial if its lattice of submodules is a series, and a hoop R is uni-serial if either RR and RR are uni-serial modules. the category of uni-serial jewelry comprises commutative valuation jewelry and closed lower than homomorphic photos. however it isn't really closed lower than direct sums nor with admire to Morita equivalence: a matrix ring over a uni-serial ring isn't uni-serial. there's a category of jewelry that is very with reference to uni-serial yet closed lower than the structures simply pointed out: serial earrings. a hoop R is termed serial if RR and RR is a right away sum (necessarily finite) of uni-serial modules. among others this type comprises triangular matrix jewelry over a skew box. additionally if F is a finite box of attribute p and G is a finite staff with a cyclic basic p-Sylow subgroup, then the gang ring FG is serial.

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2 [13, proof of Th. 2] Let P be a semi-prime Goldie ideal of a serial ring Rand ei fJ. P for every i . Then C(P) is a (right and left) Ore set . G. Puninski, Serial Rings © Kluwer Academic Publishers 2001 41 42 SERIAL RINGS Proof. Let a E R, t E T = C(P). 8) it has the semisimple Artinian classical quotient ring. So the image of T in R/ P is an Ore set, hence au = tb + p for some u E T, s e Rand pEP. 1 p = tc for certain c E P, therefore au = t(b + c). 0 Thus the previous proposition proves that we can localize at C(P) in a serial ring R, but the kernel of the corresponding homomorphism may be nonzero .

Clearly f = 7[ M t'rr' cM, and set 9 = f- I 7[M'" : K, EB K j --+ M . Then 9 is a split epimorphism (for h = 7[' ~M : M --+ K , EB K j satisfies gh = 1M) and M' EB N' = K; EB Kj, where M' = 7['~M(M) ~ M and N' = ker(g). 8) . 15 [34, Pr. 3] Let M, N I , ... , Ni be uni-serial modules over an arbitrary ring, 1 ~ 2, M is isomorphic to a direct summand of N I EB ... EB Ni and M '1 N k for every k . Then there are indices i i= i, the monomorphisms f : M --+ Ni, 9 : N; --+ M (which are not epi) and the epimorphisms (not mono) u : M --+ N j , v : N j --+ M.

Since e, ¢ Pi and the ideal Pi is prime, it is sufficient to prove that eJ(Xp )ei ~ Pi . Let us show by induction on m that ei1m(Xp )ei ~ Pi. The case m = 0 is obvious. Let ei1m(Xp)ei ~ Pi and fix an element r E Im+dXp). Then r can be written as r = Lh 8hrh8~, where Sh, rh, s~ E R, and for every h there is Xh E Xp such that either Xhrh E Im(X p) or rhxh E Im(X p) . ejrhek8~ei E Pi since P is a localizable system. Similarly, if Pk = 0 then ei8hejrhekS~ei E Pi. If P j, P k =1= 0, let us consider the condition: either Xhrh E Im(X p) or rhxh E Im(Xp).

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