A cute non-UFD ring

                You can quickly read about UFD here. (The author "Herstein", mentioned in the paragraph following Definition 4, seems to be I. N. HERSTEIN - Topics in Algebra - JOHN WILEY & SONS, 1975, Theorem at page 148)

               Let the ring $R=\mathbb{Z}_8[X]$. Units in $R$ are $\{\hat{1},\hat{3},\hat{5},\hat{7}\}$.

          The polynomial $X^2-\hat{1}$ admits two decompositions into irreducible factors:

$$X^2-\hat{1}\;=\;(X-\hat{1})(X+\hat{1})\;=\;(X-\hat{3})(X+\hat{3}). \tag{1}$$

The factor $X-\hat{1}$ is not associated with any of the factors $X\pm \hat{3}$. Neither does the other one $X+\hat{1}$. Indeed, $\hat{3}\cdot (X-\hat{1})=\hat{3}X-\hat{3}\neq X\pm \hat{3},\;\hat{5}\cdot (X-\hat{1})=-\hat{3}X+\hat{3}\neq X\pm\hat{3},\;$

$\hat{7}\cdot (X-\hat{1})=-X+\hat{1}\neq X\pm \hat{3}.$

          It can also be seen from (1) that the polynomial $X^2-\hat{1}$, although of degree two, has four roots in $\mathbb{Z}_8$.

               The example is taken from Michael ARTIN's book "ALGEBRA" (PRENTICE HALL, 1991) page 392, the example following Proposition (1.8).