When does commutativity in each homogenous component of a unital graded ring forces commutativity of whole ring?

Question

Let $R=\oplus_{g \in G} R_g$ be a graded unital ring , graded by a monoid $G$ . Suppose $x_gy_g=y_gx_g , \forall x_g , y_g \in R_g ; \forall g \in G$ ; then is it true that $R$ is commutative ? If not true in general , then what other conditions on $G$ or $R$ will force such behaviour ? Is it true when $G=\mathbb Z$ or $\mathbb N \cup \{0\}$ ? Has these questions been considered in literature ?

Answer 1

No.

Let $R$ be the ring of upper triangular $2\times 2$ matrices over a field, let $G=\mathbb{Z}$, let $R_0$ be the set of diagonal matrices, and let $R_1$ be the subset of $R$ consisting of matrices with zero diagonal.

Answer 2

There are easy counterexamples.

1) Associative case:

Take the basis $x_1,y_2,z_3,w_3$ and graded law $x_1^2=0$, $x_1y_2=z_3$, $y_2x_1=w_3$. Associativity is straightforward (the only equality of degree 3 to check is $x_1^2x_1=x_1x_1^2$).

2) Lie algebra case: the basis $x_0,y_1$ with law $x_0y_1=-y_1x_0=y_1$.