Abstract Algebra Answers

An element of R is called idempotent if a ²= a. find all idempotents of Z₆ x Z₁₂

An element of R is called idempotent if a ²=a. Show that set of all idempotents in a commutative ring is closed under multiplication.

An element of R is called idempotent if a ²=a. Show that a division ring contains exactly 2 idempotent elements.

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Does { a + b √ 2 | a, b ∈ Z } form a ring (with the usual operations in R )?

Is it commutative? Does it have a unity? Be sure to justify your answers.

Discuss Diophantine equations and expound on how it will really be valuable to students

Show that U(8)/is isomorphic to U(12)

If the pair of cycles a=(a₁,a₂,.......a_m) and b=(b₁,b₂,.......b_m) have no entries in common .show that ab =ba.

Let a be an element of.order n in a group and k be a positive integer .prove that |a^k| = n/(gcd (n,k))