Conjugacy of Group Elements and Equivalence Relations

Show that conjugacy of group elements is an equivalence relation.

Conjugacy Classes in S_3

What are the conjugacy classes in S_3? keywords: S3

Equivalence relations, surjective maps, partitions and fibers.

Consider any surjective map f from a set X onto another set Y. We can define a relation on X by x_1 ~ x_2 if f(x_1) = f(x_2). Check that this is an equivalence relation. Show that the associated partition of X is the partition into "fibers" f^(-1) (y) for y in Y. I would like to understand what this question is asking me a ...continues

Rings : Multiplicative Inverses and Identities

1) if a ring R has a multiplicative identity, then the multiplicative identity is unique. 2) if an element r that is in the ring R has a left multiplicative inverse r' and a right multiplicative inverse r", then r' = r".

Show that a set of matrices is a ring without an identity element.

I am given the set of infinite-by-infinite matrices with real entries that have only finitely many nonzero entries. Also, the matrix has entries a_ij, where i and j are natural numbers and there is a natural number n such that a_ij = 0 if i >= n or j >= n. I need to understand how to show that this set of matrices is a ring w ...continues

Show that a ring isomorphism has a multiplicative identity and is commutative.

Suppose φ:R --> S is a ring isomorphism. Show that R has a multiplicative identity if, and only if, S has a multiplicative identity. Show that R is commutative if, and only if, S is commutative.

Show that a set of real rational functions is a field.

NOTE: In this description, R represents the symbol for the set of real numbers. I couldn't find a way to type or copy the correct R symbol for the set of real numbers. Also, the parentheses in R(x) is used to distinguish the ring R(x) of rational functions from the ring R[x] of polynomials. Show that the set R(x) of rational ...continues

Rings and Principal Ideals : Left and Right Ideals

Show that if R is a ring with identity element and x is an element in R, then Rx = {rx: r in R} is the principal left ideal generated by x. Similarly, xR = {xr: r in R} is the principal right ideal generated by x.

Rings : Ideals

Let S be a subset of a set X. Let R be the ring of real-valued functions on X, and let I be the set of real-valued functions on X whose restriction to S is zero. Show that I is an ideal in R.

Show that a nonzero homomorphism of a simple ring is injective.

Show that a nonzero homomorphism of a simple ring is injective. In particular, a nonzero homomorphism of a field is injective.