Unit/Unit Sets in Rings
01-20-2011
, 10:05 AM
Hey guys,
I'm a bit confused about Unit/Invertible Elements in a Ring R.
Is the definition of a unit, x if there exists a y in R such that x*y = 1?
If this is the case why is the set of units of R in Z(integer), Z* = {+1,-1} ? Is the set of units Z* just the set in which the elements can be divided by all the members of Z? Cause I looked at a few examples and it seemed like it was the case for C(complex) C* as well, which is C* = {1,-1,i,-i}.
Many Thanks
I'm a bit confused about Unit/Invertible Elements in a Ring R.
Is the definition of a unit, x if there exists a y in R such that x*y = 1?
If this is the case why is the set of units of R in Z(integer), Z* = {+1,-1} ? Is the set of units Z* just the set in which the elements can be divided by all the members of Z? Cause I looked at a few examples and it seemed like it was the case for C(complex) C* as well, which is C* = {1,-1,i,-i}.
Many Thanks
01-20-2011
, 11:54 AM
Quote:
Hey guys,
I'm a bit confused about Unit/Invertible Elements in a Ring R.
Is the definition of a unit, x if there exists a y in R such that x*y = 1?
If this is the case why is the set of units of R in Z(integer), Z* = {+1,-1} ? Is the set of units Z* just the set in which the elements can be divided by all the members of Z? Cause I looked at a few examples and it seemed like it was the case for C(complex) C* as well, which is C* = {1,-1,i,-i}.
Many Thanks
I'm a bit confused about Unit/Invertible Elements in a Ring R.
Is the definition of a unit, x if there exists a y in R such that x*y = 1?
If this is the case why is the set of units of R in Z(integer), Z* = {+1,-1} ? Is the set of units Z* just the set in which the elements can be divided by all the members of Z? Cause I looked at a few examples and it seemed like it was the case for C(complex) C* as well, which is C* = {1,-1,i,-i}.
Many Thanks
Consider the set 2Z = the set of even integers.
This is a ring, and 2 divides everything, but 2 is not a unit. But it is true that a unit will divide every element in the ring (by definition of divisibility -- a divides b if there exists c such that a*c = b). You should prove all the statements here just to make sure you understand what's going on.
01-20-2011
, 12:00 PM
Quote:
Hey guys,
I'm a bit confused about Unit/Invertible Elements in a Ring R.
Is the definition of a unit, x if there exists a y in R such that x*y = 1? Yes
If this is the case why is the set of units of R in Z(integer), Z* = {+1,-1} ? The only divisors of 1 in Z are -1 and +1
Is the set of units Z* just the set in which the elements can be divided by all the members of Z? No, any element of a ring R can be divided by an element of the group of units R*: if z is in R and uv=1, z=z(1)=zuv so u and v divide z
Cause I looked at a few examples and it seemed like it was the case for C(complex) C* as well, which is C* = {1,-1,i,-i}. Here, you mean C={ a+bi: a,b are elements of Z }. If C is any field such as the complex numbers, C* are all the elements except for 0=the additive identity.
Many Thanks
I'm a bit confused about Unit/Invertible Elements in a Ring R.
Is the definition of a unit, x if there exists a y in R such that x*y = 1? Yes
If this is the case why is the set of units of R in Z(integer), Z* = {+1,-1} ? The only divisors of 1 in Z are -1 and +1
Is the set of units Z* just the set in which the elements can be divided by all the members of Z? No, any element of a ring R can be divided by an element of the group of units R*: if z is in R and uv=1, z=z(1)=zuv so u and v divide z
Cause I looked at a few examples and it seemed like it was the case for C(complex) C* as well, which is C* = {1,-1,i,-i}. Here, you mean C={ a+bi: a,b are elements of Z }. If C is any field such as the complex numbers, C* are all the elements except for 0=the additive identity.
Many Thanks