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By Hartmut Laue

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1). 9 Definition. Let K be a commutative unitary ring, (P, +, ·) an associative commutative unitary K-algebra. Suppose that there exists an independent subset T of the commutative monoid (P, ·) the multiplicative closure of which is a K-basis of P . Then P is called a polynomial ring over K in the set of variables (or indeterminates) T , and T is called a K-algebra basis of P . Clearly, the independence of T means that the multiplicative closure of T is a free commutative monoid over T , hence the K-algebra isomorphism P ∼ = KT • holds.

While this observation clearly is due to the particular choice of π, the polynomial in (c) is not only a fixed point for π ¯ but in regard to any chosen permutation of {1, 2, 3}. A polynomial f ∈ K[t1 , . . , tn ] is called symmetric if f π ¯ = f for all π ∈ Sn . We write Ksym [t1 , . . , tn ] for the set of all symmetric polynomials in K[t1 , . . , tn ]. It is easily seen to be a subalgebra of K[t1 , . . 5. Let n ∈ N, f ∈ Ksym [t1 , . . , tn ], B a commutative associative unitary K-algebra.

B1 , . . , bn ) → f (b1 , . . , bn ), are the polynomial functions of B n into B which, of course, are no homomorphisms in general. Thus each polynomial f ∈ P induces a polynomial function. 41 Let P be a polynomial ring over K in n variables, b1 , . . , bn (not necessarily distinct) elements of a commutative associative unitary K-algebra B. We put K[b1 , . . 3, is easily seen to be a unital K-subalgebra of B. ,bn = {f |f ∈ P, f (b1 , . . , bn ) = 0B } is a subalgebra42 of P . The n-tuple (b1 , .

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