On the product in the power series ring

Let $$A_n \colon= K[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$$ be a power series ring over a field $$K$$ in $$2n$$ variables and $${\frak m}_{A_n}$$ be the unique maximal ideal of $$A_n$$.

Suppose we have two elements $$\alpha = X_1^{e_1} \cdots X_n^{e_n}$$, $$\beta = Y_1^{f_1} \cdots Y_n^{f_n}$$ such that some of $$e_i$$ and some of $$f_i$$ are bigger than $$0$$.

We add $$A \in {\frak m}_{A_n}$$ to $$\alpha$$ and $$B \in {\frak m}_{A_n}$$ to $$\beta$$ such that $$A$$ (resp. $$B$$) comprises scalar times monomials different from $$\alpha$$ (resp. $$\beta$$).

Consequently, $$A + \alpha \in {\frak m}_{A_n}$$ (resp. $$B + \beta \in {\frak m}_{A_n}$$) contains $$\alpha$$ (resp. $$\beta$$) as its constituent.

Q. Does the multiplication $$(A + \alpha)(B + \beta)$$ always contain nonzero monomial $$X_1^{g_1} \cdots X_n^{g_n}Y_1^{g_{n+1}} \cdots Y_n^{g_{2n}}$$ up to scalar coefficient such that inequalities $$g_1 \leq 2e_1$$,...,$$g_n \leq 2e_n$$, $$g_{n+1} \leq 2f_1$$, ..., $$g_{2n} \leq 2f_n$$ hold simultaneously?

• What happens if you take the initial terms in degree lexicographic order? – Zach Teitler Jun 3 at 4:23
• What do you mean by taking the initial term? For example, you mean $\alpha = X_1^e, \beta = Y_1^f$ ? – Rinmyaku Jun 3 at 8:34