By K Heiner Kamps, Timothy Porter

Summary homotopy thought relies at the statement that analogues of a lot of topological homotopy idea and easy homotopy idea exist in lots of different different types, corresponding to areas over a hard and fast base, groupoids, chain complexes and module different types. learning express models of homotopy constitution, resembling cylinders and direction area structures allows not just a unified improvement of many examples of recognized homotopy theories, but additionally finds the interior operating of the classical spatial concept, basically indicating the logical interdependence of houses (in specific the lifestyles of sure Kan fillers in linked cubical units) and effects (Puppe sequences, Vogt's lemma, Dold's Theorem on fibre homotopy equivalences, and homotopy coherence idea)

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**Example text**

As a typical example let us mention the {first} Weyl algebra Ad~), defined as Ad~) = ~[x ][y; d~] or, alte rna tively, as EXAMPLE Al (~) = ~(x, y) / (yx - xy - 1). 6. The ring S = R[x; u, 8] satisfies the following universal property: if ljJ : R -- T is a ring homomorphism and if there exists some y E T with the property that PROPOSITION yljJ(r) = ljJ(u(r))y + 1jJ(8(r)), for all r E R, then there exists a unique ring homomorphism p : S = R[x; u, 8] -- T: x ...... 3. Let S = R [x; u, 8] be an Ore extension of R, associated to the quasiderivation (u, 8).

Qsqs+l =R and Ra ~ ... ~ RqsqS+l ~ RqS+l = R. It is easy to see that these inclusions are actually strict. ,. Pr+l = RPi+l ... Pr+l. for example, then PiPi+l ... Pr+l and Pi+l ... Pr+l are equal up to a unit. This would imply Pi to be a unit - a contradiction! 36 1. GENERALITIES ON RINGS Put Mi = Rpi ... Pr+l/Ra resp. Nj = Rqj ... qs+l/Ra, then we obtain two chains o = MI resp. C Mz C ... C Mi C ... C Mr+l =M o= NI C Nz C ... C Ni C '" C Ns+I = M, where M = R / Ra. These chains are actually composition series for M.

In order to modify the meaning of "unique" in the previous definition, let us introduce the following concept. 9. Two elements r, SER are said to be left similar, and we denote this by r ~ s, if there is an isomorphism R/ Rr ~ R/ Rs of left R -modules. DEFINITION Clearly, r ~ s implies that r ~ s, but not conversely. 8, we have t - 1 ~ t - i, for example, while t - 1 +t - i. 10. If R is commutative and if r, sER, then the following assertions are equivalent: PROPOSITION (1) (2) r r ~ ~ s; s. is obvious that the first statement implies the second one.