First off, the "R" in an R-module, is some ring. A set with an associative commutative addition with an identity and inverses which also has an associate multiplication with identity which distribute over the addition. That is:
1) (x + y) + z = x + (y + z)
2) x + y = y + x
3) 0 + x = x
4) (-x) + x = 0
5) (x·y)·z = x·(y·z)
6) 1·x = x = x·1
7) x ·(y + z) = x·y + x·z
8) (y + z)·x = y·x + z·x
For any x,y,z ∈ R. In other words, an Abelian group with a multiplication. This is a generalisation of our idea of integers. Do you know other examples of rings?
Then M is also an Abelian group with a multiplication, but only by elements of R. So a module over a ring is a set with an associative commutative addition and a (left) multiplication by elements of the ring. That is:
1) (a + b) + c = a + (b + c)
2) a + b = b + a
3) 0 + a = a
4) (-a) + a = 0
5) (r·s)·a = r·(s·a)
6) 1·a = a
7) r ·(a + b) = r·a + r·b
8) (r + s)·a = r·a+ r·s
For any a,b,c ∈ M and r,s ∈ R.
So you are just being asked to verify those 8 properties, but since addition and multiplication are both component-wise, these are really just the same 8 properties which are given when you say that R is a ring. Can you explain why?
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First off, the "R" in an R-module, is some ring. A set with an associative commutative addition with an identity and inverses which also has an associate multiplication with identity which distribute over the addition. That is:
1) (x + y) + z = x + (y + z)
2) x + y = y + x
3) 0 + x = x
4) (-x) + x = 0
5) (x·y)·z = x·(y·z)
6) 1·x = x = x·1
7) x ·(y + z) = x·y + x·z
8) (y + z)·x = y·x + z·x
For any x,y,z ∈ R. In other words, an Abelian group with a multiplication. This is a generalisation of our idea of integers. Do you know other examples of rings?
Then M is also an Abelian group with a multiplication, but only by elements of R. So a module over a ring is a set with an associative commutative addition and a (left) multiplication by elements of the ring. That is:
1) (a + b) + c = a + (b + c)
2) a + b = b + a
3) 0 + a = a
4) (-a) + a = 0
5) (r·s)·a = r·(s·a)
6) 1·a = a
7) r ·(a + b) = r·a + r·b
8) (r + s)·a = r·a+ r·s
For any a,b,c ∈ M and r,s ∈ R.
So you are just being asked to verify those 8 properties, but since addition and multiplication are both component-wise, these are really just the same 8 properties which are given when you say that R is a ring. Can you explain why?