# Different definitions of graded rings

In Atiyah i recently read the definition of a graded ring as a ring that can be written as $R=\displaystyle \bigoplus_{i \geq 0}^{\infty}R_i$ where each $R_i$ is an abelian subgroup of $R$ (with the condition that $R_iR_j \subseteq R_{i+j}$.

However I find this confusing, because in other textbooks I read definitions with $R=\displaystyle \bigoplus_{i= -\infty}^{\infty}R_i$. Why does one definition use negative index but the other doesnt? I also stumbled upon words like "$\mathbb{Z}$-grading" and "standard grading", can anyone explain this?

• @user26857, so for example if I consider $A=k[x_1,...,x_n]$ I always think that the grading of $A$ looks like $\bigoplus_{i=0}^{\infty}A_i$ where each $A_i$ consist of homogeneous polynomials of degree $i$, your point is that I might aswell think about this as a $\mathbb{Z}$-grading, by letting $A_i =0$ for all $i < 0$?
– user117449
Feb 3, 2015 at 12:27
• @user26857, Why not...well this was my fist thought aswell. But if its so easy, then any ring graded by $i=0,1,2...$ is also graded by $i \in \mathbb{Z}$ so why would atiyah make the definition of a graded ring by positive index, and not right away make the definition by also having negative index?
– user117449
Feb 3, 2015 at 12:35
• @Sodan sometimes doing things in full generality (at least at first) can hide the underlying motivation. Feb 3, 2015 at 13:01

In general you can take the grading of a ring to be any monoid $M$. This means that the ring $R$ can be written as a direct sum $R = \bigoplus_{x \in M} R_x$, where each $R_x$ is an abelian subgroup of $R$, and $R_x R_y \subseteq R_{x \cdot y}$ (where $\cdot$ is the multiplication in $M$). Such a ring is called an "$M$-graded ring".
In the first case Atiyah considers ring graded by $\mathbb{N} = \{0, 1, \dots\}$, but you can also find rings graded by $\mathbb{Z}$ as in your second example. Any ring graded by $\mathbb N$ is also graded by $\mathbb Z$, by the way, by letting $R_i = 0$ for $i < 0$.