By Angelika Wörz-Busekros (auth.)

The objective of those notes is to offer a slightly whole presentation of the mathematical conception of algebras in genetics and to debate intimately many functions to concrete genetic occasions. traditionally, the topic has its foundation in different papers of Etherington in 1939- 1941. basic contributions were given by means of Schafer, Gonshor, Holgate, Reiers¢l, Heuch, and Abraham. in the meanwhile there exist approximately 40 papers during this box, one survey article by way of Monique Bertrand from 1966 in response to 4 papers of Etherington, a paper via Schafer and Gonshor's first paper. additionally Ballonoff within the 3rd component of his publication "Genetics and Social constitution" has integrated 4 papers by way of Etherington and Reiers¢l's paper. it seems that a whole evaluation, in par ticular one comprising more moderen effects was once missing, and it was once tough for college students to go into this box of analysis. i began to write down those notes in spring 1978. a primary german model was once entire on the finish of that yr. extra revision and translation required one other yr. i am hoping that the notes of their current kingdom offer an inexpensive assessment and they will facilitate entry to this box. i'm specially thankful to Professor okay. -P. Hadeler and Professor P. Holgate for interpreting the manuscript and giving crucial reviews to all models of the textual content. i'm additionally very thankful to Dr. I. Heuch for plenty of discussions in the course of and after his remain in TUbingen. I desire to thank Dr. V. M.

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**Example text**

For this purpose we use the following idea whiqh can be found in a book [76J by Jacobson. For the given field R let F = R(~1' ••. '~n) be the field of rational functions in n (associative and commutative) indeterminates over R and let OC IF = OC®RF be the algebra over the extension IF of :the field IK. Now choose a fixed basis a 1 , ••• ,an in (l. a. is one element of F • In this context Jacobson [76J calls i=1 )~ ~ x a generic element of We apply. the ideas of the preceding section with F instead of R.

26) defines a mapping of I«X) x K(X) into K(X). Since ~ is linear this mapping is bilinear and hence defines a multiplication in K(X). We have the following theorem. 9. 26) is a commutative, associative algebra which is isomorphic with the ordinary polynomial algebra K[X] in one indeterminate over K. Proof. •. of K(X) • Thus the multipli~ation has these properties in general. Then the isomorphism is trivial. 0 After this insight has been obtained it is obvious that the Euclidian algorithm and the unique factorization of a polynomial into prime factors can be carried over to the algebra K(X) of polynomials in right prinCipal powers.

Suppose statement (2) holds. Then the algebra Ol together with the Lie algebra ':I- - of linear transformations of ot satisfy the hypothesis of a theorem of Lie, cf. 50. Thus there is a basis a o ' a 1 ' ... , an of ct with respect to which all elements of "+ are represented by lower triangular matrices. Since the ideal tt of oc. is nilpotent by assumption, tl* is also nilpotent by a theorem of Albert [8J. Therefore to all elements of~~ with respect to this basis correspond nilpotent lower triangular matrices.