Definition 4

*Let $A, A'$ be two graded rings of the same type $\Delta$ and $(A_\lambda), (A'_\lambda)$ their respective graduations. A ring homomorphism $h : A \to A'$ is called graded if $h(A_\lambda) \subset A'_\lambda$ for all $\lambda \in \Delta$.

Let $M, M'$ be two graded modules of type $\Delta$ over a graded ring of type $\Delta$. Let $u : M \to M'$ be an $A$-homomorphism and $\delta$ an element of $\Delta$; $u$ is called graded of degree $\delta$ if $u(M_\lambda) \subset M_{\lambda + \delta}$ for all $\lambda \in \Delta$.

Let $A$ be a graded ring of type $\Delta$, $A'$ a graded ring of type $\Delta'$ and $\rho : \Delta \to \Delta'$ a homomorphism. A ring homomorphism $h : A \to A'$ is called graded if $h$ is a graded homomorphism of graded rings of type $\Delta'$ when $A$ is given the graduation of type $\Delta'$ derived from its graduation of type $\Delta$ by means of $\rho$ (no. 1, Example 2); this therefore means that $h(A_\lambda) \subset A'_{\rho(\lambda)}$ for all $\lambda \in \Delta$.

An $A$-homomorphism $u : M \to M'$ is called graded if there exists $\delta \in \Delta$ such that $u$ is graded of degree $\delta$. If $u \neq 0$ and every element of $\Delta$ is cancellable, the degree $\delta$ of $u$ is then determined uniquely.

If $h : A \to A', h' : A' \to A''$ are two graded homomorphisms of graded rings of type $\Delta$, so is $h' \circ h : A \to A''$; for a mapping $h : A \to A'$ to be a graded ring isomorphism, it is necessary and sufficient that $h$ be bijective and that $h$ and the inverse mapping $h'$ be graded homomorphisms; it also suffices for this that $h$ be a bijective graded homomorphism. Thus it is seen that graded homomorphisms can be taken as the morphisms of the species of graded ring structure of type $\Delta$ (Set Theory, IV, § 2, no. 1).

Similarly, if $u : M \to M'$ and $u' : M' \to M''$ are two graded homomorphisms of graded A-modules of type $\Delta$, of respective degrees $\delta$ and $\delta'$, $u' \circ u : M \to M''$ is a graded homomorphism of degree $\delta + \delta'$. If $\delta$ admits an inverse $-\delta$ in $\Delta$ and $u : M \to M'$ is a bijective graded homomorphism of degree $\delta$, the inverse mapping $u' : M' \to M$ is a bijective graded homomorphism of degree $-\delta$. It follows as above that the graded homomorphisms of degree 0 can be taken as the morphisms of the species of graded A-module of type $\Delta$. But a bijective graded homomorphism $u : M \to N$ of degree $\neq 0$ is not a graded A-module isomorphism if $M$ and $N$ are non-zero and the elements of $\Delta$ are cancellable.