Metamath Proof Explorer


Table of Contents - 10.8.2. Ring of integers

According to Wikipedia ("Integer", 25-May-2019, https://en.wikipedia.org/wiki/Integer) "The integers form a unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring. This universal property, namely to be an initial object in the category of [unital] rings, characterizes the ring ." In set.mm, there was no explicit definition for the ring of integers until June 2019, but it was denoted by , the field of complex numbers restricted to the integers. In zringring it is shown that this restriction is a ring (it is actually a principal ideal ring as shown in zringlpir), and zringbas shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as Definition df-zring of the ring of integers.

Remark: Instead of using the symbol "ZZrng" analogous to used for the field of complex numbers, we have chosen the version with an "i" to indicate that the ring of integers is a unital ring, see also Wikipedia ("Rng (algebra)", 9-Jun-2019, https://en.wikipedia.org/wiki/Rng_(algebra)).

  1. czring
  2. df-zring
  3. zringcrng
  4. zringring
  5. zringrng
  6. zringabl
  7. zringgrp
  8. zringbas
  9. zringplusg
  10. zringsub
  11. zringmulg
  12. zringmulr
  13. zring0
  14. zring1
  15. zringnzr
  16. dvdsrzring
  17. zringlpirlem1
  18. zringlpirlem2
  19. zringlpirlem3
  20. zringinvg
  21. zringunit
  22. zringlpir
  23. zringndrg
  24. zringcyg
  25. zringsubgval
  26. zringmpg
  27. prmirredlem
  28. dfprm2
  29. prmirred
  30. expghm
  31. mulgghm2
  32. mulgrhm
  33. mulgrhm2
  34. irinitoringc
  35. nzerooringczr
  36. Example for a condition for a non-unital ring to be unital
    1. pzriprnglem1
    2. pzriprnglem2
    3. pzriprnglem3
    4. pzriprnglem4
    5. pzriprnglem5
    6. pzriprnglem6
    7. pzriprnglem7
    8. pzriprnglem8
    9. pzriprnglem9
    10. pzriprnglem10
    11. pzriprnglem11
    12. pzriprnglem12
    13. pzriprnglem13
    14. pzriprnglem14
    15. pzriprngALT
    16. pzriprng1ALT
    17. pzriprng
    18. pzriprng1