1. If I is an ideal of the ring R, show how to make the quotient...

1. If I is an ideal of the ring R, show how to make the quotient ring R/I into a left R-module, and also show how to make R/I into a right R-module.

Expert Solution

Colby Messinger answered 1 year ago

If I is an ideal of the ring R, show how to make the quotient ring...

If I is an ideal of the ring R, show how to make the quotient ring R/I into a left R-module, and also show how to make R/I into a right R-module.

For an arbitrary ring R, prove that a) If I is an ideal of R, then...

For an arbitrary ring R, prove that a) If I is an ideal of R, then I[ x] forms an ideal of the polynomial ring R[ x]. b) If R and R' are isomorphic rings, then R[ x] is isomorphic to R' [ x ].

Let R be a commutative ring with unity. If I is a prime ideal of R,...

Let R be a commutative ring with unity. If I is a prime ideal of R, prove that I[x] is a prime ideal of R[x].

Let R be a commutative ring with identity with the property that every ideal in R...

Let R be a commutative ring with identity with the property that every ideal in R is principal. Prove that every homomorphic image of R has the same property.

Show that the quotient ring Q[x]/(x2 − 3) is isomorphic to a subfield of the real...

Show that the quotient ring Q[x]/(x2 − 3) is isomorphic to a subfield of the real numbers R.

If R is a division ring, show that cent R forms a field.

6.4.13. If R is the ring of Gaussian integers, show that Q(R) is isomorphic to the...

6.4.13. If R is the ring of Gaussian integers, show that Q(R) is isomorphic to the subfield of C consisting of complex numbers with rational real and imaginary parts.

Show that R-1(a)R(a) = I, where I is the identity matrix and R(a) is the rotation matrix.

Show that R-1(a)R(a) = I, where I is the identity matrix and R(a) is the rotation matrix. This equation shows that the inverse coordinate transformation returns you to the original coordinate system.

Let R be a commutative domain, and let I be a prime ideal of R. (i)...

Let R be a commutative domain, and let I be a prime ideal of R. (i) Show that S defined as R \ I (the complement of I in R) is multiplicatively closed. (ii) By (i), we can construct the ring R1 = S-1R, as in the course. Let D = R / I. Show that the ideal of R1 generated by I, that is, IR1, is maximal, and R1 / I1R is isomorphic to the field of fractions of...

Let R be a ring and f : M −→ N a morphism of left R-modules. Show that:

Let R be a ring and f : M −→ N a morphism of left R-modules. Show that: c) K := {m ∈ M | f(m) = 0} satisfies the Universal Property of Kernels. d) N/f(M) satisfies the Universal Property of Cokernels. Q2. Show that ZQ :a) contains no minimal Z-submodule

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