I read somewhere that:

Since Z8 and Z30 both have identities, we know that the ideals in R = Z8 ⊕Z30

all have the form I ⊕ J where I is an ideal in Z8 and J is an ideal in Z30. In order for I ⊕ J

to be maximal, one of I or J must be maximal and the other must be the entire ring.

I understand to be maximal one of I or J must be maximal. Why should the other one be the entire ring?