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By G. Puninski

the most subject matter in classical ring idea is the constitution conception of jewelry of a specific sort. for instance, nobody textual content ebook in ring idea may perhaps pass over the Wedderburn-Artin theorem, which says ring R is semisimple Artinian iffR is isomorphic to a finite direct sum of complete matrix earrings over skew fields. this is often an instance of a finiteness which, at the least traditionally, has ruled in ring thought. Ifwe want to give some thought to a demand of a lattice-theoretical kind, except being Artinian or Noetherian, the main common is uni-seriality. the following a module M is termed 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 earrings and closed below homomorphic photos. however it isn't closed below 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 as regards to uni-serial yet closed lower than the structures simply pointed out: serial earrings. a hoop R is named serial if RR and RR is a right away sum (necessarily finite) of uni-serial modules. among others this classification contains triangular matrix earrings over a skew box. additionally if F is a finite box of attribute p and G is a finite team with a cyclic general p-Sylow subgroup, then the crowd ring FG is serial.

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Serial Rings by G. Puninski

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