Classification of matrix idempotents over a subalgebra of $K[x]$ generated by monomials

Let $K$ be a commutative integral domain such that all finitely generated projective modules over $K[x]$ are free. Let $\Lambda$ be a subalgebra of $K[x]$ generated by monomials, such that there are only finitely many monomials in $K[x]$ which do not belong to $\Lambda$.
For such algebras, the following results are obtained: matrix idempotents over $\Lambda$ are described up to conjugation; provided that $\frac12\in K$, finite-dimensional representations of a group of order 2 over $\Lambda$ are described up to an isomorphism; and all finitely generated projective $\Lambda$-modules are described up to an isomorphism.
These results can be generalized to the case of subalgebras of the algebra of polynomials $K[x_1,\dots,X_n]$ with $n>1$.
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