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Thank you for giving me the opportunity to assist you. I will give the best answer that I can with the information provided.
This is not true at all. Generally the EIC claim goes with the dependency claim, but there are special rules for children of divorced and separated parents.
Presuming that you and the ex lived apart in 2010, and your son lived with you for more than half the year, you are considered the custodial parent under IRS rules and your son is your qualifying child. You may release the dependency exemption to the other parent. The non-custodial parent may claim the child as a dependent and claim the child tax credit for that child, but cannot use that child to claim head of household status, child and dependency tax credit, or earned income credit.
You as the custodial parent are allowed to use the child for head of household status, earned income credit, and child and dependent tax credit if applicable.
The fact that your ex is claiming the child as a dependent does not preclude you from claiming these benefits, and the fact that you are claiming these benefits does not preclude your ex from claiming your son as a dependent.
Please let me know if you need additional information or references.
So the non-custodial parent cannot claim the EITC, correct? Also, do you have a reference for that?
That is correct.
The following is a link to the EIC publication. It has all of the rules and a lot of examples, some similar to your situation, and discusses rules for children of divorced or separated parents on page 19:
The specific paragraph the summarizes what I have told you:
If a child is treated as the qualifying child of the noncustodial parent under the special rule just described for children of divorced or separated parents or parents who live apart, only the noncustodial parent can claim an exemption and the child tax credit for the child.
However, the custodial parent, if eligible, or another eligible taxpayer can claim the child
as a qualifying child for the EIC and other tax benefits listed on page 16. If the child is the
qualifying child of more than one person for these benefits, then the tiebreaker rules
determine which person can treat the child as a qualifying child.