Master Submodules: Enhance Your Understanding of Abstract Algebra

Question 1

In the context of ring theory, for a ring R and an R-module M, what condition must be met for N to be considered a submodule of M?

Accepted Answer

N is closed under addition and scalar multiplication.

Answer

N is closed under addition.

Answer

N is closed under scalar multiplication.

Question 2

Which of the following accurately defines a submodule over a ring R?

Accepted Answer

A non-empty subset of a module that remains closed under both addition and scalar multiplication operations

Answer

A module existing as a subgroup within a given module

Answer

A quotient module arising from a particular subset

Answer

A subring of the ring R

Question 3

Given a module M over a ring R and a subset N of M, which condition ensures that N qualifies as a submodule?

Accepted Answer

Closure under both addition and scalar multiplication operations

Answer

Closure under either addition or scalar multiplication operation

Answer

Non-emptiness of N

Answer

N being a subgroup of M

Question 4

Let M be a module over a ring R, and N and K be submodules of M. Which of the following is always a submodule of M?

Accepted Answer

The intersection of N and K

Answer

The sum of N and K

Answer

The union of N and K

Answer

The quotient module N/K

Question 5

In the context of a module M over a ring R, what is the definition of a quotient module?

Accepted Answer

A module resulting from dividing M by a submodule

Answer

A subset of M that forms a module under the same operations as M

Answer

A homomorphism from M to another module

Answer

A ring that is isomorphic to M

Question 6

Regarding modules M and N over a ring R, which statement about the quotient module M/N is accurate?

Accepted Answer

Its elements are equivalence classes of elements from M

Answer

It is always isomorphic to M

Answer

It is always isomorphic to N

Answer

Its elements are submodules of M

Question 7

Given a module M over a ring R and a submodule N of M, which statement is always true?

Accepted Answer

The intersection of M and N is a submodule of M

Answer

The union of M and N is a submodule of M

Answer

The quotient module M/N is always a free module

Answer

The quotient module N/M is always a cyclic module

Question 8

Given a module M over a ring R and a submodule N of M, which of the following is NOT necessarily a submodule of M?

Accepted Answer

The set of elements of M that are not in N

Answer

N + N

Answer

rN for any scalar r in R

Answer

The set-theoretic intersection of N with any other submodule of M

Question 9

Let M be a module over a ring R and let N be a submodule of M. Which of the following is isomorphic to the quotient module M/N?

Accepted Answer

The set of all cosets of N in M.

Answer

The set of all homomorphic images of N in M.

Answer

The set of all direct sums of N with any other submodules of M.

Answer

The set of all submodules of M that contain N.

Question 10

Let M be a module over a ring R. Which of the following is equivalent to saying that M is a simple module?

Accepted Answer

M has no proper submodules.

Answer

M is isomorphic to a quotient module of a field.

Answer

M is a direct sum of cyclic modules.

Answer

M is a free module.

Question 11

Let M be a module over a ring R and let N be a submodule of M. Which of the following statements is true?

Accepted Answer

M/N is a module over R.

Answer

N is a module over R.

Answer

M/N is a ring.

Answer

N is a ring.

Question 12

Let M be a module over a ring R and let N be a submodule of M. Which of the following is a necessary and sufficient condition for N to be a direct summand of M?

Accepted Answer

N is a projective module.

Answer

N is a free module.

Answer

N is an injective module.

Answer

N is a flat module.

Question 13

Let M be a module over a ring R. Which of the following is a necessary and sufficient condition for M to be a free module?

Accepted Answer

M has a basis.

Answer

M is a direct sum of cyclic modules.

Answer

M is a projective module.

Answer

M is an injective module.

Question 14

For a module M over a ring R, the quotient module formed by a submodule N, written as M/N, is defined as

Accepted Answer

{m + N | m ∈ M}

Answer

{m - N | m ∈ M}

Answer

{m/N | m ∈ M}

Answer

{N/m | m ∈ M}

Question 15

Which of the following is NOT an example of a submodule?

Accepted Answer

Functions from set X to set Y

Answer

Even integers within the module of integers

Answer

Zero-determinant matrices within the module of matrices

Answer

Polynomials of degree at most n within the module of polynomials

Question 16

Which of the following accurately describes a submodule of a module?

Accepted Answer

A non-empty subset closed under addition and scalar multiplication

Answer

A proper subset containing the zero vector

Answer

A subset closed under scalar multiplication but not addition

Answer

A subset isomorphic to the original module

Question 17

If H is a submodule of M, then M/H is a:

Accepted Answer

Factor module

Answer

Module

Answer

Vector space

Answer

Ring

Question 18

Which of the following is a property of minimal submodules?

Accepted Answer

They are the smallest non-zero submodules.

Answer

They are the largest submodules.

Answer

They are always cyclic.

Answer

They are always direct sums of cyclic modules.

Question 19

If H and K are submodules of M, then the intersection of H and K is a:

Accepted Answer

Submodule

Answer

Module

Answer

Vector space

Answer

Ring

Question 20

If M is a finitely generated module over a principal ideal domain R, then every submodule of M is:

Accepted Answer

Finitely generated

Answer

Free

Answer

Cyclic

Answer

Principal

Question 21

If M is a module over a ring R and N is a submodule of M, then the annihilator of N in R is:

Accepted Answer

The set of all elements in R that annihilate N.

Answer

The set of all elements in R that annihilate M.

Answer

The set of all elements in M that annihilate N.

Answer

The set of all elements in N that annihilate M.

Question 22

Consider a module M over a ring R and a submodule N of M. Which of the following sets is NOT a submodule of M?

Accepted Answer

The set of inverses of elements in N.

Answer

The set of scalar multiples of elements in N.

Answer

The set of sums of elements in N.

Answer

The set of differences of elements in N.

Question 23

Given a module M over a ring R and a submodule N, what is a factor module of M?

Accepted Answer

M / N

Answer

N / M

Answer

M x N

Answer

Hom(M, N)

Question 24

Which property is characteristic of quotient modules?

Accepted Answer

They are always modules.

Answer

They are always submodules of the original module.

Answer

They are always isomorphic to the original module.

Answer

They are always free modules.

Question 25

Let M be a module over a ring R and N a submodule of M. What condition is necessary and sufficient for M to be a free module?

Accepted Answer

N is a direct summand of M.

Answer

N is a maximal submodule of M.

Answer

N is a minimal submodule of M.

Answer

N is a cyclic submodule of M.

Question 26

Let M be a module over a ring R and N a submodule of M. What condition is necessary and sufficient for M to be a projective module?

Accepted Answer

Every submodule of M is a direct summand of M.

Answer

Every submodule of M is a maximal submodule of M.

Answer

Every submodule of M is a minimal submodule of M.

Answer

Every submodule of M is a cyclic submodule of M.

Question 27

Which of the following is a necessary and sufficient condition for a non-empty subset S of a module M to be a submodule?

Accepted Answer

S is closed under addition and scalar multiplication, and it contains the zero vector.

Answer

S is closed under addition and scalar multiplication.

Answer

S is closed under addition and contains the zero vector.

Answer

S is closed under scalar multiplication and contains the zero vector.

Question 28

Let M be a module over a ring R. If S is a submodule of M, then the quotient module M/S is

Accepted Answer

a module over R, but not necessarily a free module.

Answer

not a module over R.

Answer

a vector space over R, but not necessarily a free module.

Answer

a free module over R.

Question 29

Which of the following is NOT a property of submodules?

Accepted Answer

They are always direct summands.

Answer

They are closed under addition.

Answer

They are closed under scalar multiplication.

Answer

They are invariant under module homomorphisms.

Question 30

Let M be a module over a ring R and let N be a submodule of M. Which of the following is equivalent to saying that N is a maximal submodule of M?

Accepted Answer

There is no other submodule of M that properly contains N.

Answer

N is the largest submodule of M.

Answer

N is a submodule of every other submodule of M.

Answer

N is the unique maximal submodule of M.

Question 31

What must a subset of a module satisfy to be considered a submodule?

Accepted Answer

It is closed under scalar multiplication.

Accepted Answer

It inherits the same operations as the module.

Answer

It is always a proper subset of the module.

Answer

It is always linearly independent.

Question 32

Which of the following statements about submodules is false?

Accepted Answer

Submodules may not be closed under scalar multiplication.

Answer

Submodules are closed under addition.

Answer

Submodules always contain the zero element of the module.

Question 33

Consider the module Z_6. Which of the following subsets qualifies as a submodule of Z_6?

Accepted Answer

{0, 2, 4}

Answer

{1, 2, 3, 4}

Answer

{1, 3, 5}

Answer

{0, 1, 2, 3}

Question 34

Let M be a module over a ring R, with N as a submodule of M. Which statement is equivalent to N being a maximal submodule of M?

Accepted Answer

There exists no other submodule of M that properly contains N.

Answer

N is the smallest submodule within M.

Answer

N stands as the largest submodule within M.

Question 35

Let M be a module over a ring R, with N as a submodule of M. What is the nature of the intersection of all submodules of M that contain N?

Accepted Answer

It represents the smallest submodule of M that encapsulates N.

Answer

It is identical to N.

Answer

It is equivalent to the zero submodule.

Answer

It represents the largest submodule of M that encapsulates N.

Question 36

Given a submodule N of a module M, what is the quotient module M/N?

Accepted Answer

The set of all equivalence classes of elements of M under the equivalence relation defined by N.

Answer

The set generated by the differences of elements in M and N.

Answer

The set of all elements of M that are not in N.

Answer

The set of all cosets of N in M that have a representative in a given set.

Question 37

Which of the following is true about the intersection of two submodules?

Accepted Answer

It is a submodule that is contained in both of the original submodules.

Answer

It is always the zero submodule.

Answer

It is always a proper submodule.

Answer

It is only a submodule if the two submodules are the same.

Question 38

Let M be a module and N a submodule of M. Which of the following is NOT an element of the annihilator of N?

Accepted Answer

Any element of M that is not in N.

Answer

0

Answer

Any element of M that annihilates every element of N.

Answer

Any element of M that is annihilated by every element of N.

Question 39

Let M be a module and N a submodule of M. Which of the following is true about the natural homomorphism from M to M/N?

Accepted Answer

It is surjective and has kernel N.

Answer

It is injective.

Answer

It is an isomorphism.

Answer

It is neither surjective nor injective.

Question 40

Let M be a module over a ring R. If S is a submodule of M, then the quotient module M/S is:

Accepted Answer

The set of all cosets of S in M.

Answer

The set of all submodules of S.

Answer

The set of all homomorphisms from S to M.

Question 41

Let M be a module over a ring R. If S is a submodule of M, then the annihilator of S in R is:

Accepted Answer

The set of all elements of R that annihilate every element of S.

Answer

The set of all elements of M that annihilate S.

Answer

The set of all ideals of R that contain S.

Answer

The set of all submodules of M that contain S.

Question 42

Which of the following is a necessary and sufficient condition for a submodule to be a direct summand of a module?

Accepted Answer

It is isomorphic to a quotient module of the module.

Answer

It is a free module.

Answer

It is a projective module.

Question 43

Let M be a module over a ring R. If S is a submodule of M, then the factor module M/S is:

Accepted Answer

A module over R.

Answer

An ideal of R.

Answer

A vector space over the same field as M.

Answer

A submodule of M.

Question 44

Which of the following statements is true regarding submodules?

Accepted Answer

The intersection of any collection of submodules is always a submodule.

Answer

Every submodule is a proper subset of the module.

Answer

The union of any collection of submodules is always a submodule.

Question 45

Given a module M and its submodule N, what does the factor module M/N represent?

Accepted Answer

The set of all cosets of N within M.

Answer

The set of all elements in M that are not in N.

Answer

The set of all scalar multiples of N.

Answer

The set of all submodules of M that contain N.

Question 46

For a ring R and a module M over R, which of these is always a submodule of M?

Accepted Answer

The set of all elements in M that are annihilated by a specific element of R.

Answer

The set of all units of R.

Answer

The set of all zero divisors of R.

Question 47

What condition guarantees that a submodule N is a direct summand of the module M?

Accepted Answer

There exists a submodule K of M such that M is the direct sum of N and K (M = N ⊕ K).

Answer

N has the same cardinality as M.

Answer

N is a free module.

Question 48

Consider the module M consisting of all functions mapping a set A to a ring R. Which of these is a submodule of M?

Accepted Answer

The set of functions that equal zero at a specific element of A.

Answer

The set of all surjective functions.

Answer

The set of all invertible functions.

Question 49

Given a module M over a ring R and a submodule N of M, which statement is **always** true?

Accepted Answer

N is also an R-module.

Answer

N is a vector space over R.

Answer

N is an ideal of R.

Answer

N is a ring.

Question 50

For a ring R and an R-module M, which of the following sets is **not** necessarily a submodule of M?

Accepted Answer

The set of all elements in M that are not divisible by a fixed element r in R.

Answer

The intersection of two submodules of M.

Answer

The set of all scalar multiples of a fixed element m in M.

Question 51

Which of the following is a submodule of the module Z over the ring Z?

Accepted Answer

The set of all even integers.

Answer

The set of all prime numbers.

Answer

The set of all odd integers.

Answer

The set of all positive integers.

Question 52

Given a module M over a ring R and a submodule N, what is the quotient module M/N?

Accepted Answer

The set of all cosets of N in M, where addition and scalar multiplication are defined using representatives.

Answer

The set of all elements in M that are not in N.

Answer

The set of all submodules of M that contain N.

Question 53

Let M be a module over a ring R, and let N be a submodule of M. Which of the following statements is **true** regarding the canonical homomorphism from M to M/N?

Accepted Answer

It is a surjective homomorphism with kernel N.

Answer

It is an injective homomorphism with kernel N.

Answer

It is an isomorphism.

Question 54

Let R be a ring and M an R-module. Which of the following conditions is **sufficient** to guarantee that a subset S of M is a submodule?

Accepted Answer

S is closed under addition and scalar multiplication by elements of R.

Answer

S is closed under scalar multiplication by elements of R.

Answer

S contains the zero element of M.

Question 55

Consider the module R[x] over the ring R. Which of the following is a submodule of R[x]?

Accepted Answer

The set of all polynomials of degree less than or equal to n, where n is a fixed non-negative integer.

Answer

The set of all polynomials with constant term 1.

Question 56

Let M be a module over a ring R. Which of the following statements is **false**?

Accepted Answer

Every submodule of M is also a quotient module of M.

Answer

The zero submodule is a submodule of M.

Answer

The intersection of two submodules is a submodule.