Dummit And Foote Solutions Chapter 7 __link__ -

Chapter 7 of Abstract Algebra by David S. Dummit and Richard M. Foote marks a pivotal transition in the standard undergraduate algebra curriculum. Moving away from the group theory covered in Chapters 1 through 6, this chapter introduces the fundamental definitions and properties of .

"Find all subrings of ( \mathbbZ \times \mathbbZ )." Solution narrative: Subrings must be additive subgroups of ( \mathbbZ^2 ), so they are of the form ( m\mathbbZ \times n\mathbbZ ) or diagonal-like sets? Wait, additive subgroups of ℤ² are all ( (a,b) : a,b \in \mathbbZ, (a,b) \text satisfies some linear condition ). But closure under multiplication restricts them. You'll find the only subrings are ( m\mathbbZ \times n\mathbbZ ) and ( (a,ka): a\in\mathbbZ ) etc. This is a classic exercise—check that the diagonal set is indeed closed under multiplication: ((a,ka)(b,kb) = (ab, k^2 ab)) — that's not of the form ((x,kx)) unless ( k^2 = k ) (so ( k=0 ) or 1). Good insight! dummit and foote solutions chapter 7

You discover that not every ring has a multiplicative identity (1). And even if it does, a subring might not contain it. Chapter 7 of Abstract Algebra by David S

Use the Subring Test . Check if it is closed under subtraction and multiplication, and contains Moving away from the group theory covered in