3 - STRUCTURE OF NONCOMMUTATIVE RINGS

Summary

If we use the standard identification of the complex number a + bi. with the ordered pair (a, b) in the Euclidean plane R², then multiplication in C allows us to multiply ordered pairs. This multiplication respects the structure of R² as a vector space over R, where r(a, b) = (ra, rb) for r ∈ R, since if x = (a, b) and y = (c, d), then r(xy) = (rx)y = x(ry). Hamilton discovered the quaternions H (recall Example 1.1.6) in 1843 after all of his efforts to find such a multiplication on R³ ended in failure. He finally realized that it was necessary to work in R⁴ and that he had to drop the requirement of commutativity. In our current terminology, Hamilton had found the most elementary example of a division ring. (Recall Definition 1.1.6: a division ring satisfies all of the axioms of a field, with the possible exception of the commutative law for multiplication.) We note that Wedderburn showed in 1909 that it is impossible to give a finite example of a division ring that is not a field.

An algebra over a field F is a ring A that is also a vector space over F, in which c(xy) = (cx)y = x(cy) for all x, y ∈ A and c ∈ F. The algebra A is called finite dimensional if A is a finite dimensional vector space over F, and it is called a division algebra if it is a division ring. (Refer to Definition 1.5.7.)