Question

2. Show that the ring of all 2 imes 2 matrices egin{pmatrix} a & b \ 0 & c end{pmatrix} such that a is an integer and b, c are rational is right Noetherian but not left Noetherian. 3. Show that the ring of all 2 imes 2 matrices egin{pmatrix} a & b \ 0 & c end{pmatrix} such that a is rational and b, c are real is right Artinian but not left Artinian. 4. Show that a nonzero module A has a composition series if and only if A satisfies both the ascending and descending chain conditions on submodules. Deduce now that if D is a division ring, then the ring Mat(n, D) of all n imes n matrices over D is both Artinian and Noetherian.

          2. Show that the ring of all 2 	imes 2 matrices egin{pmatrix} a & b \ 0 & c end{pmatrix} such that a is an integer and b, c are rational is right Noetherian but not left Noetherian.
3. Show that the ring of all 2 	imes 2 matrices egin{pmatrix} a & b \ 0 & c end{pmatrix} such that a is rational and b, c are real is right Artinian but not left Artinian.
4. Show that a nonzero module A has a composition series if and only if A satisfies both the ascending and descending chain conditions on submodules. Deduce now that if D is a division ring, then the ring Mat(n, D) of all n 	imes n matrices over D is both Artinian and Noetherian.

2. Show that the ring of all 2 imes 2 matrices eginpmatrix a b 0 c endpmatrix such that a is an integer and b, c are rational is right Noetherian but not left Noetherian.
3. Show that the ring of all 2 imes 2 matrices eginpmatrix a b 0 c endpmatrix such that a is rational and b, c are real is right Artinian but not left Artinian.
4. Show that a nonzero module A has a composition series if and only if A satisfies both the ascending and descending chain conditions on submodules. Deduce now that if D is a division ring, then the ring Mat(n, D) of all n imes n matrices over D is both Artinian and Noetherian.

Added by Christopher V.

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