HW7 (Chapter 12, Problem 2), MA453, Fall 2008, Prof. Walther
I was wandering is the unity of the ring the same as the identity and for this problem is the unity the same for multiplication and addition I am a little confused? Nate Shafer
I think so. If you take the Cayley table of {0,2,4,6,8} under multiplication mod 10, you will find that when you multiply 6 by a number mod 10, you get that number (ex. 6x2=2x6=12mod10=2.) So 6 is the unity (or identity) because when you multiply 6 by a number it does not change. --Neely Misner
So are you saying that 6 is also the unity under addition? I do not understand this part.
I'm not sure about the addition part, but the element '6' checks by the unity element properties section of Theorem 12.1 on page 237. -Tim F
It's my understanding that "unity" is the identity only under multiplication. It wouldn't make sense to call an additive identity a unity. Example: 0 might be the identity under addition for some ring, but then the unity, i.e. identity for that same ring under multiplication, might be 1.
However, you can have a ring like {0} in which the unity is equal to the additive identity, which is zero.
