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.

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- Nov 11th 2006, 01:50 PMSultanOfSwingUnits 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. - Nov 11th 2006, 01:59 PMThePerfectHacker
The units in $\displaystyle \mathbb{Z}$ are $\displaystyle -1,1$

Thus,

$\displaystyle (-1,1),(-1,-1),(1,-1),(1,1)$

Quote:

(ii) F(e), where F is a field and e^2=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.

$\displaystyle \det (A)\not = 0$