Could someone point me to a proof which shows that an algebra over a ring can be presented as a quotient of a polynomial ring (in possibly infinitely many variables).
Algebra over a ring
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commutative-algebra
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0Never mind, I figured it out. – 2011-07-24
1 Answers
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Let $A$ be an $R$ algebra. Let $X$ be a set of variables $x_a$ which arein bijection with the set $A$. Consider the unique map of $R$-algebras $f:R[X]\to A$ which maps $x_a$ to $a$ for all $a\in A$. This is clearly surjective, so $A\cong R[X]/\ker f$.
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1@user3714: notice that my construction uses many, many more variables than what is ever needed. Is is enough to use one variable per element in a generting set of $A$ as an $R$-algebra. – 2011-07-24