Homogeneous elements of graded algebras as polynomials of algebra generators

Question

Let $A = \oplus_{i > 0} A_{i}$ be a positively graded $k-$algebra and let $\mathcal{B} = \left\{b_{1}, \ldots, b_{n}\right\}$ be a generating set of $A$. Let $f$ be a homogeneous element in $A_{i}$ and so $f$ can be written as polynomial in the generators $$f = \sum_{\alpha \in \mathbb{N}^{n}} \lambda_{\alpha}b^{\alpha}$$ where $\lambda_{\alpha}$ is in $k$ and $b^{\alpha} = b_{1}^{\alpha_{1}}b_{2}^{\alpha_{2}} \cdots b_{n}^{\alpha_{n}}$. Suppose that $\lambda_{\alpha} \neq 0$ then must it be the case that $b^{\alpha}$ is a homogeneous element in $A_{i}$?

Answer 1

Assume that $k$ is a field with $\text{char}\, k \neq 2$ and let $A = k[x,y]$ with $\deg x \neq \deg y$. Then $\{x-y, x+y\}$ forms a generating set for $A$. The element $f = (x+y) + (x-y)$ is homogeneous, but neither "monomial" in the generators is homogeneous.