What is an Image in Abstract Algebra?

[bwinter]

My professor can't give me a straight answer, the word is absent from the appendix of the book, and google search returns nothing.

So my question is, in the context of abstract algebra...

what the heck is an image?

For example...my book says "Note that the image of the unity is the unity of the image but not the unity of Z₃₀."

What does that mean?

 

[Bill Simpson]

Perhaps this will help you. http://en.wikipedia.org/wiki/Image_(mathematics )

 

[bwinter]

Thanks very much for the quick reply. I'm very surprised that didn't come up in my internet search. Seriously, try searching "image abstract algebra" and see what nonsense comes up. Anyway...

So f(x) is the image of x?

So the image of the unity, is f(1).

The unity of the image is the value of x so that f(x) = 1?

And the unity of Z₃₀ is the elements of Z₃₀ under addition such that when one is added to an element of Z30, the same element is returned? Wouldn't that just be {0, 30}?

Please confirm my assumptions here so I know which way is up.

 

[Office_Shredder]

Under addition, the image of the unity is f(0)?

I don't think I've ever seen unity to refer to zero, only the multiplicative identity

The unity of the image is the value of x so that f(x) = 1 (under multiplication) and f(x) = 0 (under addition)?

It would help if you tell us exactly what the function is. Certainly f(x) can't be both 1 and 0 at the same time though

 

[bwinter]

Sorry, you're right. I get unity and identity blurred together sometimes. The multiplicative identity 1. I edited the post to clarify.

The function is the mapping from Z₅ to Z₃₀ given by x --> 6x.

 

[Office_Shredder]

So what's happening here is that in Z₃₀, the subset {0,6,12,18,24} (the image of the map) is a ring, with 6 the multiplicative identity. But 6 is not the multiplicative identity of Z₃₀