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11

Expert-verifiedFound in: Page 94

Book edition
3rd

Author(s)
Thomas W Hungerford, David Leep

Pages
608 pages

ISBN
9781111569624

**Show that 1+3x is a unit in ${\mathrm{\mathbb{Z}}}_{9}\left[x\right]$ . Hence, Corollary 4.5 may be false if R is not an integral domain.**

It is proved that 1+3x is a unit in ${\mathrm{\mathbb{Z}}}_{9}\left[x\right]$ .

If any given function R[x] is a ring, then the commutative, associative, and distributive laws hold such that the function f(x)+g(x) exists.

** **

The given unit is 1+3x . The multiplicative inverse of this will be: 1-3x.

Then, we have:

$\left(1+3x\right)\left(1-3x\right)=1-9{x}^{2}=1$

Clearly found to be in ${\mathrm{\mathbb{Z}}}_{9}\left[x\right]$ .

Hence proved, 1+3x is a unit in ${\mathrm{\mathbb{Z}}}_{9}\left[x\right]$ .

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