Subring Test - Profile By Commutative Subrings

Profile By Commutative Subrings

A ring may be profiled by the variety of commutative subrings that it hosts:

• The quaternion ring H contains only the complex plane as a planar subring
• The coquaternion ring contains three types of commutative planar subrings: the dual number plane, the split-complex number plane, as well as the ordinary complex plane
• The ring of 3 × 3 real matrices also contains 3-dimensional commutative subrings generated by the identity matrix and a nilpotent ε of order 3 (εεε = 0 ≠ εε). For instance, the Heisenberg group can be realized as the join of the groups of units of two of these nilpotent-generated subrings of 3 × 3 matrices.