  Various authors have been generalizing some unital ring properties to
nonunital rings. We consider properties related to cancellation of modules
(being unit-regular, having stable range one, being directly finite, exchange,
or clean) and their "local" versions. We explore their relationships and extend
the defined concepts to graded rings. With graded clean and graded exchange
rings suitably defined, we study how these properties behave under the
formation of graded matrix rings. We exhibit properties of a graph $E$ which
are equivalent to the unital Leavitt path algebra $L_K(E)$ being graded clean.
We also exhibit some graph properties which are necessary and some which are
sufficient for $L_K(E)$ to be graded exchange.
