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In 1914, E. Cartan posed the matter of discovering all irreducible actual linear Lie algebras. Iwahori gave an up to date exposition of Cartan's paintings in 1959. This concept reduces the class of irreducible genuine representations of a true Lie algebra to an outline of the so-called self-conjugate irreducible complicated representations of this algebra and to the calculation of an invariant of any such illustration (with values $+1$ or $-1$) known as the index.

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Amer. Math. Soc. 110, 196–212 (1964). [8] Dech´ene, L. I. Adjacent Extensions Of Rings; Ph. D. thesis; University of California at Riverside: Riverside, CA, 1978. [9] Dobbs, D. E. Internat. J. Commut. Rings 1, 173–179 (2002). [10] Dobbs, D. E. Comm. Algebra 34, 3875–3881 (2006). [11] Dobbs, D. E. Comm. Algebra 35, 773–779 (2007). [12] Dobbs, D. E. Comm. Algebra 37, 604–608 (2009). [13] Dobbs, D. ; and Picavet-L’Hermitte, M. Comm. Algebra 33, 3091–3119 (2005). [14] Dobbs, D. ; Picavet-L’Hermitte, M.

9] that the above definitions of R, S, T give an example of type (2, 3) with the asserted properties. 4 in the next section. 17, one may well ask what can be concluded in general about the ring extension S ⊂ ST when one is given distinct composable minimal ring extensions R ⊂ S and R ⊂ T . 18 summarizes what is known in this regard. As usual, if E is a vector space over a field k, then dimk (E) denotes the k-vector space dimension of E. 3]). Let R be a ring. Let R ⊂ S and R ⊂ T be distinct integral minimal ring extensions with the same crucial maximal ideal M.

The other cases are handled by ad hoc constructions. Their favor is typified by the following construction, which works for the case a = 2, b = 3. Let F be an arbitrary field, with X and Y (as usual) algebraically independent indeterminates overF, and define R := F[{X n (X 2 − X ), X n(X 2 − X )Y, X sY 2 , X tY 3 | n, s,t ≥ 0}]. The fact that R is Noetherian can be seen most easily via Eakin’s Theorem (cf. [25, Exercise 15, page 54]), which applies since F[X ,Y ] is a module-finite ring extension of R.