equalizer in Ring

Proposition

Let 20240128-equalizer_in_ring_ed8d003b66e6574aa9fb9345deda6426c5a39a2b.svg be rings and

20240128-equalizer_in_ring_6deb3591fbf00fa087a4b53456112b5cf92e253c.svg

Then the equalizer exists and is given by

20240128-equalizer_in_ring_02a942e62edf9810603d6fddf3d282ff4b0870a0.svg

with the natural inclusion 20240128-equalizer_in_ring_182347000392dbaeed23d82f1d471eac849e8350.svg

Proof

Note that 20240128-equalizer_in_ring_9b431eb75dce9ab6a85f5920816397dc8b70ddcd.svg and 20240128-equalizer_in_ring_42cc3ab66165c2291f85fe0fde0a841cac2c7fe5.svg

additive closed

Suppose 20240128-equalizer_in_ring_5f2ba5cdc3d96b53345289967f495c6058c51e8e.svg such that 20240128-equalizer_in_ring_d901b7ba3242d3f0a40174bd02505a06e9064890.svg .
Then

20240128-equalizer_in_ring_5e55114a2f60a3e239ac05332eed04db9f482482.svg

multiplicative closed

Suppose 20240128-equalizer_in_ring_5f2ba5cdc3d96b53345289967f495c6058c51e8e.svg such that 20240128-equalizer_in_ring_d901b7ba3242d3f0a40174bd02505a06e9064890.svg .
Then it follows, that

20240128-equalizer_in_ring_14e0f5d29f6d4b9d744c99be9ccb14ed6c8f4940.svg

and hence 20240128-equalizer_in_ring_b7b01c30a55eef7f157e9cd7d9d93a3bc0f07341.svg