When 2*16 congruent to 5*29 do we still have Unique Factors?

Commutative algebra / abstract algebra was a joy to study, and coupled with Number theory is a superb grounding for second and third year undergraduate study.

Current lecture notes and set texts seem to suggest that whenever p is prime, that Z/pZ is an integral domain, a unique factors setup, ... and a Field.

Quoting from some MIT lecture notes:
"We call R a Unique Factorization Domain(UFD) if every nonzero element is a product of irreducible elements in a unique way up to order and units"

Take a look at the image above and see what you think?

The key thing that is easy to overlook is the part of the definition that says 'irreducible elements'

Here we would need to check each of the 4 elements 2,16,5,29 are irreducible mod 113.
We would not say that 16 is 'irreducible'

Links and References:

• https://www.quora.com/Why-do-integers-mod-a-prime-number-create-a-finite-field
• MIT lecture notes - commutative algebra
• List of supplementary commutative algebra texts - mathoverflow list