Units of Ring

• November 11th 2006, 01:50 PM
SultanOfSwing
Units of Ring
New to the board and neep help:

Find the units of the ring R when R is
(i) Z x Z
(ii) F(e), where F is a field and e^2=0.
(iii) M2(F), where F is a field and M2 is a two-by-two matrix.

Thank you.
• November 11th 2006, 01:59 PM
ThePerfectHacker
Quote:

Originally Posted by SultanOfSwing
New to the board and neep help:

Find the units of the ring R when R is
(i) Z x Z

The units in $\mathbb{Z}$ are $-1,1$
Thus,
$(-1,1),(-1,-1),(1,-1),(1,1)$
Quote:

(ii) F(e), where F is a field and e^2=0.
This is called a trivial ring, or the ring where $1=0$
The additive element is the only unit.

Unless the notion F(e) means something else?
(It cannot possibly means a simple extension).
Quote:

(iii) M2(F), where F is a field and M2 is a two-by-two matrix.
All invertible matrices, that is,
$\det (A)\not = 0$