3
votes
1answer
82 views

If $\mathbb{Z}$ satisfies an identity $\eta$, then every **commutative** ring satisfies $\eta$? And related questions.

Assume all rings have unity and that ring homomorphisms preserve unity. Now by general principles, if every free object in the category of rings satisfies an identity $\eta$, then every object in the ...
-2
votes
1answer
163 views

presentation of the direct sum of commutative rings / algebras

If $I,J$ are index sets, $R$ a commutative unital ring, $\mathfrak{a},\mathfrak{b}$ ideals of polynomial rings $R[x_i; i\!\in\!I]$, $R[y_j; j\!\in\!J]$, and $\langle\langle\ldots\rangle\rangle$ is the ...