Rings
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What is the additive inverse of each element in the ring R described in Example 3.4?
Verify that the subset {u,v} of the ring R in Example 3.4 is a subring. Show that after a change of notation this is the same ring as described in Example 3.5.
For the ring R described in Example 3.4, use the tables to verify each of the following:
a) (u+v) + w = u + (v+w)
b) (v+w) + x = v + (w+x)
c) w(v+x) = wv + wr
d) (w + v)x = wx + vr
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Solution Summary
This provides examples of finding additive inverse, verifying a subset is a subring, and verifying statements.
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(2) According to the table, u acts as the additive identity, hence if an element z of this set has an additive inverse, z', it must satisfy z + z' = u.
Since u + u = u, u is its own (additive) inverse.
Similarly, v + v = u, so v is its own inverse.
In fact, you'll notice that every element is self-inverse (since there are u's going down the diagonal of the
addition ...
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