- #1
coquelicot
- 299
- 67
Let ##A## be a ring and ##B## be a commutative algebra over A·
Suppose that ##B## is generated by algebraic elements ##\beta\in B## over ##A##, meaning that ##\beta## fulfils a relation of the form ##P(\beta)=0##, with ##P\in A[X]##.
Is ##B## necessarily associative ?
NOTE: As usual, ##\beta^i## is defined inductively by ##\beta^i = \beta^{i-1}\beta##.
By commutativity, it follows that ##\beta\beta^i = \beta^i \beta = \beta^{i+1}##, hence by an evident induction, ##\beta^i\beta^j = \beta^{i+j}##.
I think that the answer to the question is YES because of the following reason:
We can suppose without loss of generality that ##B## is generated by a finite number of algebraic elements over ##A##.
Suppose first ##B=A[\beta]## (that is, ##B## is generated by 1 element over ##A##).
We have ##(\beta^i\beta^j)\beta^k = \beta^{i+j+k}=\beta^i(\beta^j\beta^k)##, hence using the Cauchy product, for polynomials in ##\beta## over ##A##, it follows that ##B## is associative.
If now ##B## has a finite number of algebraic generators, the result is true by an evident induction.
Suppose that ##B## is generated by algebraic elements ##\beta\in B## over ##A##, meaning that ##\beta## fulfils a relation of the form ##P(\beta)=0##, with ##P\in A[X]##.
Is ##B## necessarily associative ?
NOTE: As usual, ##\beta^i## is defined inductively by ##\beta^i = \beta^{i-1}\beta##.
By commutativity, it follows that ##\beta\beta^i = \beta^i \beta = \beta^{i+1}##, hence by an evident induction, ##\beta^i\beta^j = \beta^{i+j}##.
I think that the answer to the question is YES because of the following reason:
We can suppose without loss of generality that ##B## is generated by a finite number of algebraic elements over ##A##.
Suppose first ##B=A[\beta]## (that is, ##B## is generated by 1 element over ##A##).
We have ##(\beta^i\beta^j)\beta^k = \beta^{i+j+k}=\beta^i(\beta^j\beta^k)##, hence using the Cauchy product, for polynomials in ##\beta## over ##A##, it follows that ##B## is associative.
If now ##B## has a finite number of algebraic generators, the result is true by an evident induction.
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