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[Sorry for answering my own question, and the more so because this is happening for the second time in 24 hours.]

The question might be open. In fact, a positive answer would imply an equally positive answer to a question stated in the introduction of Melvin Henriksen's paper

"Rings with a unique regular element", pp. 78-87 in Rings, modules and radicals (Hobart, 1987), Pitman Res. Notes Math. Ser., 204, Longman Sci. Tech., Harlow, 1989,

where Henriksen writes:

We do not know if there is a ring with a unique regular ring [sic] that fails to be commutative.

In Henriksen's paper, a ring need not be unital; and a regular element is nothing else than a cancellative element of the multiplicative semigroup of the ring (loc. cit., Definition 2.1).

The question is marked as open by David Feldman in a 2012 post from the "Not especially famous, long-open problems which anyone can understand" big list (see also the comments under the same post), where he writes:

Must a non-commutative ring (with identity) contain a non-zero-divisor aside from the identity?

[Sorry for answering my own question, and the more so because this is happening for the second time in 24 hours.]

The question might be open. In fact, a positive answer would imply an equally positive answer to a question stated in the introduction of Melvin Henriksen's paper

• "Rings with a unique regular element", pp. 78-87 in B.J. Gardner (ed.), Rings, modules and radicals (Proc. Conf., Hobart/Aust. 1987), Pitman Res. Notes Math. Ser. 204, Longman Sci. Tech., Harlow, 1989,

where Henriksen writes:

We do not know if there is a ring with a unique regular ring [sic] that fails to be commutative.

In Henriksen's paper, a ring need not be unital; and a regular element is nothing else than a cancellative element of the multiplicative semigroup of the ring (loc. cit., Definition 2.1).

The question is marked as open by David Feldman in a 2012 post from the "Not especially famous, long-open problems which anyone can understand" big list (see also the comments under the same post), where Feldman writes:

Must a non-commutative ring (with identity) contain a non-zero-divisor aside from the identity?

[Sorry for answering my own question, and the more so because this is happening for the second time in 24 hours.]

The question might be open. In fact, a positive answer would imply an equally positive answer to a question stated in the introduction of Melvin Henriksen's paper

"Rings with a unique regular element", pp. 78-87 in Rings, modules and radicals (Hobart, 1987), Pitman Res. Notes Math. Ser., 204, Longman Sci. Tech., Harlow, 1989,

where Henriksen writes:

We do not know if there is a ring with a unique regular ring [sic] that fails to be commutative.

In Henriksen's paper, a ring need not be unital; and a regular element is nothing else than a cancellative element of the multiplicative semigroup of the ring.

It seems the question is marked as open by David Feldman in a post from the "Not especially famous, long-open problems which anyone can understand" big list (see also the comments under the same post). And I say "seems" because I'm not sure to understand David Feldman's formulation of the problem, as he writes:

Must a non-commutative ring (with identity) contain a non-zero-divisor aside from the identity?

"Aside from" means "in addition to" (or does it?), and so I presume that the word "identity" refers to the additive identity (that is, zero). But then, isn't the question in this formulation equivalent to asking whether there exists a non-commutative unital ring whose unique zero divisor is zero (the answer to which is obviously yes)? I must be missing something.

[Sorry for answering my own question, and the more so because this is happening for the second time in 24 hours.]

The question might be open. In fact, a positive answer would imply an equally positive answer to a question stated in the introduction of Melvin Henriksen's paper

"Rings with a unique regular element", pp. 78-87 in Rings, modules and radicals (Hobart, 1987), Pitman Res. Notes Math. Ser., 204, Longman Sci. Tech., Harlow, 1989,

where Henriksen writes:

We do not know if there is a ring with a unique regular ring [sic] that fails to be commutative.

In Henriksen's paper, a ring need not be unital; and a regular element is nothing else than a cancellative element of the multiplicative semigroup of the ring (loc. cit., Definition 2.1).

The question is marked as open by David Feldman in a 2012 post from the "Not especially famous, long-open problems which anyone can understand" big list (see also the comments under the same post), where he writes:

Must a non-commutative ring (with identity) contain a non-zero-divisor aside from the identity?

[Sorry for answering my own question, and the more so because this is happening for the second time in 24 hours.]

The question might be open. In fact, a positive answer would imply an equally positive answer to a question stated in the introduction of Melvin Henriksen's paper

"Rings with a unique regular element", pp. 78-87 in Rings, modules and radicals (Hobart, 1987), Pitman Res. Notes Math. Ser., 204, Longman Sci. Tech., Harlow, 1989.

It seems the question is marked as open by David Feldman in a post (here) on the "Not especially famous, long-open problems which anyone can understand" big list (see also the comments to that post). But, I'm not sure to understand David Feldman's rendition of Henriksen's problem, as he writes:

Must a non-commutative ring (with identity) contain a non-zero-divisor aside from the identity?

Isn't this equivalent to asking whether there exists a non-commutative unital ring whose unique zero divisor is zero (the answer to which is obviously yes)?

[Sorry for answering my own question, and the more so because this is happening for the second time in 24 hours.]

The question might be open. In fact, a positive answer would imply an equally positive answer to a question stated in the introduction of Melvin Henriksen's paper

"Rings with a unique regular element", pp. 78-87 in Rings, modules and radicals (Hobart, 1987), Pitman Res. Notes Math. Ser., 204, Longman Sci. Tech., Harlow, 1989,

where Henriksen writes:

We do not know if there is a ring with a unique regular ring [sic] that fails to be commutative.

In Henriksen's paper, a ring need not be unital; and a regular element is nothing else than a cancellative element of the multiplicative semigroup of the ring.

It seems the question is marked as open by David Feldman in a post from the "Not especially famous, long-open problems which anyone can understand" big list (see also the comments under the same post). And I say "seems" because I'm not sure to understand David Feldman's formulation of the problem, as he writes:

Must a non-commutative ring (with identity) contain a non-zero-divisor aside from the identity?

"Aside from" means "in addition to" (or does it?), and so I presume that the word "identity" refers to the additive identity (that is, zero). But then, isn't the question in this formulation equivalent to asking whether there exists a non-commutative unital ring whose unique zero divisor is zero (the answer to which is obviously yes)? I must be missing something.