In this work, ascending chain conditions in free Baxter algebras are studied by making use of explicit constructions of free Baxter algebras that the author and W. Keigher gave [in Adv. Math. 150, No. 1, 117–149 (2000; Zbl 0947.16013)]. Ascending chain conditions are investigated both for ideals and for Baxter ideals. The free Baxter algebras under consideration include free Baxter algebras on sets and free Baxter algebras on algebras. Complete free Baxter algebras are also considered.
Given \(C\) a commutative ring and \(\lambda\) an element of \(C\), a Baxter \(C\)-algebra of weight \(\lambda\) is a commutative \(C\)-algebra \(R\) with a \(C\)-linear operator \(P\) that satisfies the Baxter identity \[ P(x)P(y)=P(xP(y))+ P(yP(x)) +\lambda P(xy),\quad \forall x,y\in R. \]
As a generalization of free Baxter algebras on sets, free Baxter algebras on \(C\)-algebras are defined as follows: Given \(A\) a \(C\)-algebra, a free Baxter algebra on \(A\) is a Baxter algebra \((F_C (A),P_A)\) with a \(C\)-algebra homomorphism \(j_A:A\rightarrow F_C(A)\) that satisfies the following universal property: For any Baxter \(C\)-algebra \((R,P)\) and any \(C\)-algebra homomorphism \(\varphi:A\rightarrow R\), there exists a unique Baxter \(C\)-algebra homomorphism \(\tilde{\varphi}:(F_C (A), P_A)\rightarrow (R,P)\) such that \(\tilde{\varphi} \circ j_A= \varphi\). If \(X\) is a set and \(A=C[X]\), then \(F_C(A)\) is the free Baxter algebra on \(X\) in the usual sense.
Motivated by the shuffle product of iterated integrals, an explicit description of free Baxter algebras was given in the above-mentioned paper: For \(\lambda\in C\) and \(A\) a \(C\)-algebra, the resulting free Baxter algebra on \(A\) of weight \(\lambda\) is denoted by \((III_C(A),P_A)\) and is called the shuffle Baxter \(C\)-algebra on \(A\) of weight \(\lambda\). For a given set \(X\), the corresponding shuffle Baxter \(C\)-algebra obtained by taking \(A=C[X]\), \((III_C(C[X]),P_{C[X]})\) is called the shuffle Baxter \(C\)-algebra on \(X\) (of weight \(\lambda\)) and denoted by \((III_C(X),P_X)\).
In [Adv. Math. 151, No. 1, 101–127 (2000; Zbl 0964.16027)], L. Guo and W. Keigher obtained a completion of \(III_C(A)\) in a manner similar to completing a free \(C\)-algebra and got a complete \(C\)-algebra. The obtained complete Baxter algebra is called the complete shuffle Baxter algebra on \(A\) and denoted by \((\widehat{III}_C (A),\widehat P)\).
Related to the ascending chain condition for ideals in the free Baxter \(C\)-algebra on \(A\), the author proves that, for the case when \(A=C\) (i.e. \(A=C[X]\), with \(X\) the empty set), \(III_C(C)\) is a Noetherian ring if, and only if, \(C\) is a Noetherian \(\mathbb{Q}\)-algebra. He also proves several results on \(\widehat{III}_C(C)\): (i) If \(C\) is a Noetherian \(\mathbb Q\)-algebra and if \(\lambda=0\), then \(\widehat{III}_C(C)\) is a Noetherian ring; (ii) If \(C\) is a \(\mathbb{Q}\)-algebra, \(\lambda\in C\) is a non-zero divisor and \(\bigcap_{n\in\mathbb{N}}\lambda^n C\neq 0\), then \(\widehat{III}_C(C)\) is not a Noetherian ring; (iii) If \(C\) is not a \(\mathbb{Q}\)-algebra, then \(\widehat{III}_C(C)\) is not a Noetherian ring.
For the general case, given \(C\) a ring of characteristic zero and \(X\) any non-empty set, it is proven that the free Baxter algebra \(III_C(X)\) is not a Noetherian algebra.
The author also studies the ascending chain condition for Baxter ideals. Given \((R,P)\) a Baxter \(C\)-algebra of weight \(\lambda\) with Baxter operator \(P\), a Baxter ideal of \((R,P)\) is an ideal \(I\) of \(R\) such that \(P(I)\subseteq I\). Starting for the case when \(A=C\), it is proven that if \(C\) is a Noetherian ring, then \(III_C(C)\) and \(\widehat{III}_C(C)\) have the ascending chain condition for Baxter ideals. For the general case, given \(A\) a \(C\)-algebra, consider the \(A\)-module \(A\otimes A\) with \(A\) acting on the left tensor factor and \(M\) a \(C\)-submodule of \(A\). Denote by \(\overline{A\otimes M}\) the \(A\)-submodule of \(A\otimes A\) that is the image of \(A\otimes M\) in \(A\otimes A\) under the natural map \(A\otimes M\rightarrow A\otimes A\) induced by \(M\hookrightarrow A\). Then, it is proven:
1. Let \(\mathcal{M}\) be the partially ordered set consisting of \(A\)-submodules of \(A\otimes A\) of the form \(\overline{A\otimes M}\), where \(M\) runs through the \(C\)-submodules of \(A\). If \(\mathcal{M}\) does not satisfy the ascending chain condition, then \(III_C(A)\) of weight zero does not satisfy the ascending chain condition for Baxter ideals.
2. If \(A\) is not a Noetherian ring, then \(III_C(A)\) of any weight does not satisfy the ascending chain condition for Baxter ideals.
As a corollary of the former results, it is obtained that, taking \(A=C[X]\), if \(X\) is not empty, then \(III_C(X)\) of weight zero does not have the ascending chain condition for Baxter ideals and, if \(X\) is infinite, then \(III_C(X)\) of any weight does not have the ascending chain condition for Baxter ideals (the case \(X=\emptyset\) corresponds to \(A=C\)).