In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with a·x = x·a = 1. Stated differently, a ring is a division ring if and only if the group of units is the set of all non-zero elements.
Division rings differ from fields only in that their multiplication is not required to be commutative. However, by Wedderburn's little theorem all finite division rings are commutative and therefore finite fields. Historically, division rings were sometimes referred to as fields, while fields were called “commutative fields”.
Read more about Division Ring: Relation To Fields and Linear Algebra, Examples, Ring Theorems, Related Notions
Famous quotes containing the words division and/or ring:
“The division between the useful arts and the fine arts must not be understood in too absolute a manner. In the humblest work of the craftsmen, if art is there, there is a concern for beauty, through a kind of indirect repercussion that the requirements of the creativity of the spirit exercise upon the production of an object to serve human needs.”
—Jacques Maritain (1882–1973)
“I was exceedingly interested by this phenomenon, and already felt paid for my journey. It could hardly have thrilled me more if it had taken the form of letters, or of the human face. If I had met with this ring of light while groping in this forest alone, away from any fire, I should have been still more surprised. I little thought that there was such a light shining in the darkness of the wilderness for me.”
—Henry David Thoreau (1817–1862)