I am reviewing for the mathematics GRE subject exam, and I have this review book, and in the book when they speak of the characteristic of a ring they give the following def "Let R be a ring. The smallest positive integer n such that na = 0 for every a in R is called the characteristic of the ring R, and we write char R = n. I no such n exists, then R has characteristic 0." They then ask a question "what's char Z(adsbygoogle = window.adsbygoogle || []).push({}); _{n}?" with the answer given as n. Makes sense according to the def.

However, when I went to review further in my old abstract algebra book I am given a different definition: that characteristics are only defined for rings of unity that are also integral domains and that the characteristic is the order of the unity. (for example in Z_{3}, its a commutative ring of unity with unity as 1, and the smallest number such that if you do the additive group operation on 1 that it generates the additive identity, is 3 because 1 + 1 + 1 = 3mod3=0, so this would have characteristic of 3). My book then gives an example: "what's char Z_{2}? What's char Z_{6}?" with the answer given as "char Z is zero (because order of 1 is infinite), char Z_{2}=2, and no characteristic for Z_{6}because it's not an integral domain." Also makes sense according to this new def.

I am confused now how I would answer the question about Z_{6}on the exam!

Also, if the characteristic is defined this way (order of unity), does that mean that if the order of the unity is infinite, then the order of all elements in the ring are infinite so there are no cyclic subgroups?

Thanks!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Characteristic - must be on an integral domain? Books disagree?

**Physics Forums | Science Articles, Homework Help, Discussion**