Focus on the axioms (abelian group under +, associative under
You discover that not every ring has a multiplicative identity (1). And even if it does, a subring might not contain it.
Use the Subring Test . Check if it is closed under subtraction and multiplication, and contains dummit and foote solutions chapter 7
Solving simultaneous congruences and decomposing rings into direct products. Problem-Solving Strategies
Construction of the quotient field (like Qthe rational numbers Zthe integers Focus on the axioms (abelian group under +,
"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!
This section introduces the classification of elements within a ring. Check if it is closed under subtraction and
Often used by graduate students for a more "terse" and high-level perspective.