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

[dumbQuestion]

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_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 Z₃, 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 Z₂? What's char Z₆?" with the answer given as "char Z is zero (because order of 1 is infinite), char Z₂=2, and no characteristic for Z₆ 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 Z₆ 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!

 

[DonAntonio]

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_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 Z₃, 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 Z₂? What's char Z₆?" with the answer given as "char Z is zero (because order of 1 is infinite), char Z₂=2, and no characteristic for Z₆ 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 Z₆ 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!

What is that "old abstract algebra book"? Never mind, burn it...anyway, characteristic of ring in ALMOST any decent algebra

book is defined for any unitary ring, without any condition on being integral domain, which would make this

definition ridiculous as any finite integral domain is a field...

DonAntonio

 

[dumbQuestion]

wow, thanks so much for answering this, because until now i was going to go with the def in the book thinking the review textbook was most likely the least trustworthy of the two!