Exam (elaborations) MAT3702 Assignment 2 Solutions 2020 S1 MAT3702 Assignment 2 Solutions 2020 S1 1. Let 0R, 1R ∈ R be the zero and identity elements of R respectively. We have for all z ∈ R that 0Rz = 0R = z0R so that 0R ∈ Z(R). For all x, y ∈ Z(R), we have that xr = rx, ry = yr for all r ∈ R. Thus r(x − y) = rx − ry = xr − yr = (x − y)r making x − y ∈ Z( R). Also r(xy) = (rx)y = (xr)y = x(ry) = x(yr) = (xy)r for all r ∈ R making xy ∈ Z(R). Hence Z(R) is a subring of...