Invariant Basis Number and Types for Strongly Graded Rings

For any pair (n′, k′) of positive integers we produce a strongly Z2-graded ring R such that the identity component R0 has invariant basis number while the basis-number type of R is (n′, k′). Using different methods, for any pairs of positive integers (n, k) and (n′, k′) having n′ ≤ n and k′ k we produce a strongly Z2-graded ring R such that the basis-number type of R0 is (n, k) while the basis-number type of R is (n′, k′).