Take the semigroup ring of a semigroup with two or more multiplicative right identities. For example, the semigroup $$S = \langle a, b | ab = bb = b, ba = aa = a \rangle$$ works (it is the universal example, so if it fails then no example can work). Multiplication in this semigroup can be described as follows: every word evaluates to the last letter in it. The resulting semigroup ring is literally the free ring on two not necessarily identical right identities.

Take the semigroup ring of a semigroup with two or more multiplicative right identities. For example, the semigroup $$S = \langle a, b | ab = aa = a, ba = bb = b \rangle$$ works (it is the universal example, so if it fails then no example can work). Multiplication in this semigroup can be described as follows: every word evaluates to the first letter in it. The resulting semigroup ring is literally the free ring on two not necessarily identical right identities.

Take the semigroup ring of a semigroup with two or more multiplicative right identities. For example, the semigroup $$S = \langle a, b | ab = bb = b, ba = aa = a \rangle$$ works (it is the universal example, so if it fails then no example can work). Multiplication in this semigroup can be described as follows: every word evaluates to the last letter in it.

Take the semigroup ring of a semigroup with two or more multiplicative right identities. For example, the semigroup $$S = \langle a, b | ab = bb = b, ba = aa = a \rangle$$ works (it is the universal example, so if it fails then no example can work). Multiplication in this semigroup can be described as follows: every word evaluates to the last letter in it. The resulting semigroup ring is literally the free ring on two not necessarily identical right identities.