I read in a book the following definition of $Proj$.
Given $A=\bigoplus\limits_{n\geq0}A_n$ a $\mathbb{N}$-graded $\mathbb{C}$-algebra without nilpotent, we can define $$Proj(A)= \text{homogeneous ideals of }\\A\text{ maximal among homogeneous ideals not containing the irrelevant ideal }A_+=\bigoplus\limits_{n>0}A_n$$
However, trying to look at an example, it doesn't look right. For example, let $A=\mathbb{C}[x,y]$ with the usual gradation by polynomials degree. I expect to find out that the assignment $$ \mathbb{P}^1 \ni [a:b] \mapsto (bx-ay) \subset \mathbb{C}[x,y]$$ gives a bijection between points in $\mathbb{P}^1$ and elements of $Proj(A)$. However, $(bx-ay)$ are not maximal among homogeneous ideals not containing $A_+$. In fact, we could add $\bigoplus\limits_{n>1}A_n$ to $(bx-ay)$.
Therefore, I think that the author meant $$Proj(A)= \text{homogenous ideals of }\\A\text{ maximal among homogeneous prime ideals not containing the irrelevant ideal }A_+$$