Abstract Algebra Answers

Let ℤ[𝑥] be the polynomial ring with coefficients in ℤ. Prove or disprove 

that the ideal 𝐼 = (4, 𝑥) is a principal ideal in ℤ[𝑥]. Is the ideal 𝐼 a 

maximal ideal in ℤ[𝑥]? Explain your answer.

Let Z[X] be the polynomial ring with coefficients in Z.prove or disprove that the ideal I= (4,x) is a principal ideal in Z[X]. Is this ideal I a maximal ideal in Z[x]? Explain your answer

prove that the element (u,v) in direct product r×s of two rings r and s is unit if and only if u is a unit in r and v is unit in s

Describe (list the elements, give the identity and inverses) the cyclic

group generated (under multiplication) by [ 1 1 ]

[0 1] matrix

Prove that every cyclic group is abelian. give an example to show the converse is NOT true.

Prove or disprove: the set of diagonal n × n matrices with no zeroes on the diagonal is a subgroup of

GL(n,R)

Let D be an integral domain. Then prove that there exists a 

field F that contains a subring isomorphic to D.

Show that the lines

r·(4i+3j)−1=0, r·(−2i+j)−1=0, r·(i+2j)−1=0 ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃

are concurrent.

We showed that if S and T are isomorphic groups and e is the identity for S ,φ(e) gives the identity for T. Now show that if a and a′ are inverses in S, then φ (a) and φ (a′) are inverses in T

. (In other words, show that φ (a′) = (φ(a))'

Let G be a group with operation * and identity e. Prove that if

x * x = e for all x ∈ G, then G is abelian. Hint: consider

( a * b) * (a*b )