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Products of fields are semihereditary. This follows from the facts that products of fields are von Neumann regular and that von Neumann regular rings are semiherediatry. A proof can be found (as suggested by David White) in Lam's Lectures on modules and rings, Example 2.32 d).

(In the above, fields and rings are not necessarily commutative.)

Products of fields are semihereditary. This follows from the facts that products of fields are von Neumann regular and that von Neumann regular rings are semihereditary. A proof can be found (as suggested by David White) in Lam's Lectures on modules and rings, Example 2.32 d).

(In the above, fields and rings are not necessarily commutative.)

Yes. Products of fields are semihereditary, and products of countably many finite fields are hereditary. This follows from the facts that products of fields are von Neumann regular, that von Neumann regular rings are semiherediatry, and that countable von Neumann regular rings are hereditary. Proofs can be found (as suggested by David White) in Lam's Lectures on modules and rings, Examples 2.32 d) and e).

(In the above, fields and rings are not necessarily commutative.)

Products of fields are semihereditary. This follows from the facts that products of fields are von Neumann regular and that von Neumann regular rings are semiherediatry. A proof can be found (as suggested by David White) in Lam's Lectures on modules and rings, Example 2.32 d).

(In the above, fields and rings are not necessarily commutative.)

Yes. Products of fields are semihereditary, and products of countably many countable fields are hereditary. This follows from the facts that products of fields are von Neumann regular, that von Neumann regular rings are semiherediatry, and that countable von Neumann regular rings are hereditary. Proofs can be found (as suggested by David White) in Lam's Lectures on modules and rings, Examples 2.32 d) and e).

(In the above, fields and rings are not necessarily commutative.)

Yes. Products of fields are semihereditary, and products of countably many finite fields are hereditary. This follows from the facts that products of fields are von Neumann regular, that von Neumann regular rings are semiherediatry, and that countable von Neumann regular rings are hereditary. Proofs can be found (as suggested by David White) in Lam's Lectures on modules and rings, Examples 2.32 d) and e).

(In the above, fields and rings are not necessarily commutative.)