SCHIZOPHRENIA IN CONTEMPORARY MATHEMATICS 2 7 In [l] the Gelfand theory of commutative Banach algebras was con- structivized to a certain extent. The theory has to be considered unsatis- factory, not because the classical content is not recovered (it is), but because it is so ugly. It is ahnost certain that a prettier constructivization will some- day be found. Stolzenberg [28] gives a meticulous analysis of some of the considera- tions involved in constructivizing a particular classical theory, the open map- ping theorem and related material. Again, an incisive constructivization is not obtained. J. Tennenbaum [29] gives a deep and intricate constructive version of Hilbert's basis theorem. Consider a commutative ring A with unit. It would be tempting to call A (constructively) Noetherian if for each sequence {a 3 n of elements of A there exists an integer N such that for n 2 N the element a is a linear combination of a . , a with coefficients in A. This n n- 1 notion would be worthless - - not even the ring of integers is Noetherian in this sense. In case A is discrete (meaning that the equality relation for A is decidable), the appropriate constructive version of Noetherian seems to be the following (as given in [29] ). DEFINITION. A sequence [a 1 of elements of A is almost eventu- n ally zero if for each sequence of positive integers there exists a posi- tive integer k such that a = O for k S n r k t n "k. DEFINITION. A basis operation r for A is a rule that to each finite sequence al,.. . ,an of elements of A assigns an element r(a l,...,an) of A of the form an + X a + . . + X a , where each X belongs to A. 1 1 n-1 n-1 i DEFINITION. A is Noetherian if it has a basis operation r such that for each sequence {a 1 of elements of A the associated sequence n [r (al, . . . , a ) 1 is almost eventually zero. n Tennenbaum proved the appropriateness of his version of Noetherian by checking out the standard cases and proving the Hilbert basis theorem. He also extended his definition and results to the case of a not-necessarily dis- crete ring A. The theory in that case is so complex that it cannot be consid- ered satisfactory. In spite of the pioneering efforts of Kronecker, and continued work by many algebraists, resulting in many deep theorems, the systematic
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