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In an $\Bbb{N}$-graded ring $R=\bigoplus_nR_n$, an element is called homogenous (of degree $n$) if it is contained in $R_n$. An ideal is called homogenous if it is generated by homogenous elements. Let $I$ be a homogenous ideal generated by homogenous elements $f_1,\dots,f_s$. Now it says, given $f\in I$ homogeneous, we can write $f=\sum_ig_if_i$ with $g_i$ homogeneous of degree $\deg f-\deg f_i$. No proof is given. How can I see this?

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$I$ is generated by homogeneous elements $f_1,…,f_s$. so $f=\sum_ig_if_i$. if $g_t$ were not homogeneous (for some t) use associative property ($f_i$s may be Repeated). $\deg f-\deg f_i$, by definition of graded rings ($R_i R_j \subset R_{i+j}$).
(in fact $\deg g_if_i=\deg g_i+\deg f_i = \deg f$, for all $i$)

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